analysis.bigb

= Analysis
{wiki}

= Topological analysis
{parent=Analysis}

= Brouwer fixed-point theorem
{c}
{parent=Topological analysis}

Every continuous self-map of a closed disc has a fixed point.

= Polynomial root from a disk self-map
{parent=Brouwer fixed-point theorem}

If a polynomial equation can be rearranged as $z=F(z)$ with a continuous $F$ mapping a closed disk into itself, Brouwer's theorem guarantees a root in that disk.

= Poincare-Miranda theorem
{parent=Brouwer fixed-point theorem}
{c}
{wiki=Poincaré–Miranda_theorem}

If each component of a continuous map on a box has opposite weak signs on the corresponding pair of faces, the map has a zero.

= Clamped-map proof of planar path crossing
{parent=Poincare-Miranda theorem}

Projecting a sign-corrected displacement map back to a square turns Brouwer's fixed point into a zero of the displacement, proving that paths joining opposite side pairs intersect.

= No-retraction theorem
{parent=Topological analysis}

No closed disc retracts continuously onto its boundary. This is equivalent to <Brouwer fixed-point theorem> by the standard ray construction and the antipodal map.

= Metric space
{parent=Topological analysis}
{wiki}

A metric space is a set equipped with a nonnegative symmetric distance satisfying definiteness and the triangle inequality.

= Convergence in a metric space
{title2=$x_n\to x$}
{parent=Metric space}

A sequence $(x_n)$ converges to $x$ in a metric space when $d(x_n,x)\to0$, equivalently when every neighbourhood of $x$ contains all sufficiently late terms.

= Sequential characterization of continuity in metric spaces
{parent=Metric space}

A map between metric spaces is continuous exactly when $x_n\to x$ always implies $f(x_n)\to f(x)$. If an inverse image of an open set were not open, points outside it could be chosen within distance $1/n$ of a point inside it, proving the converse by contradiction.

= Sequential compactness of a compact metric space
{parent=Metric space}

Every sequence in a compact metric space has a convergent subsequence. Conversely, sequential compactness and compactness are equivalent for metric spaces.

= Normality of every metric space
{parent=Metric space}

Every metric space is normal. For disjoint closed sets $A,B$, the function
$$
x\longmapsto\frac{d(x,A)}{d(x,A)+d(x,B)}
$$
is continuous, equals zero on $A$, and equals one on $B$; inverse images of disjoint neighbourhoods of zero and one separate the sets.

= Bounded-continuous-function characterization of compact metric spaces
{parent=Metric space}

A metric space $X$ is compact exactly when every continuous function $X\to\mathbb R$ is bounded. For the converse, a noncompact metric space contains a closed discrete sequence; the unbounded function assigning its $n$th point the value $n$ extends to $X$ by the Tietze extension theorem.

= Completeness
{parent=Metric space}
{wiki=Complete_metric_space}

A metric space is complete when every <Cauchy sequence> converges to a point of the space.

= Triangle inequality
{parent=Metric space}
{wiki}

A metric satisfies $d(x,z)\leq d(x,y)+d(y,z)$. A norm satisfies the corresponding inequality $\|u+v\|\leq\|u\|+\|v\|$.

= Euclidean distance
{parent=Metric space}
{wiki}

The Euclidean distance between $x,y\in\mathbb R^d$ is
$$
\|x-y\|_2=\sqrt{\sum_{i=1}^d(x_i-y_i)^2}.
$$

= Diameter of a metric space
{title2=$\operatorname{diam}(S)$}
{parent=Metric space}
{wiki=Diameter}

The diameter of a subset $S$ of a metric space is
$$
\operatorname{diam}(S)=\sup\{d(x,y):x,y\in S\}.
$$

= Uniform continuity
{parent=Metric space}
{wiki}

Uniform continuity requires one input tolerance to work at every point of the domain for a chosen output tolerance.

= Heine-Cantor theorem
{parent=Uniform continuity}
{c}
{wiki=Heine%E2%80%93Cantor_theorem}

Every continuous map from a compact metric space to a metric space is uniformly continuous.

= Uniform limit theorem for uniformly continuous functions
{parent=Uniform continuity}

A uniform limit of uniformly continuous maps between metric spaces is uniformly continuous. Approximate the limit at both endpoints by one fixed member of the sequence and use its uniform continuity between them.

= Equivalent metrics
{parent=Metric space}
{wiki=Equivalence_of_metrics}

Two metrics are equivalent when they induce the same topology. Global two-sided constant bounds imply equivalence, but equivalence alone need not supply such bounds.

= Equivalent metrics need not be bi-Lipschitz equivalent
{parent=Equivalent metrics}

On $\mathbb R$, the usual metric and $|\arctan x-\arctan y|$ induce the same topology, but their ratio along $(0,n)$ tends to zero. Thus no positive global lower comparison constant exists.

= Equivalent codomain metrics need not preserve uniform convergence
{parent=Equivalent metrics}

Topological equivalence does not determine uniform convergence. For the usual and arctangent metrics on $\mathbb R$, maps that change the value $n$ to $2n$ at a single domain point have arctangent sup-error tending to zero while their usual sup-error diverges.

= Distance to a set
{parent=Metric space}
{wiki=Distance_from_a_point_to_a_set}

For a metric space, $d(x,S)=\inf_{s\in S}d(x,s)$ is $1$-Lipschitz and vanishes on the closure of $S$.

= Isolated point
{parent=Metric space}
{wiki=Isolated_point}

A point $x$ of a metric space is isolated when some open ball about $x$ contains no other point of the space, equivalently when the singleton $\{x\}$ is open.

= Complete metric space
{parent=Metric space}
{wiki}

A metric space is complete when every Cauchy sequence converges to a point of the space.

= Cantor intersection theorem
{parent=Complete metric space}
{c}
{wiki=Cantor's_intersection_theorem}

A metric space is complete if and only if every decreasing sequence of nonempty closed sets whose diameters tend to zero has nonempty intersection. In that case the intersection contains exactly one point.

= Baire category theorem
{parent=Complete metric space}
{c}
{wiki}

Every complete metric space is a Baire space: every countable intersection of open dense subsets is dense. Equivalently, no nonempty open subset is a countable union of nowhere-dense subsets.

To prove the dense-intersection form, let $V$ be nonempty and open and let $G_1,G_2,\ldots$ be open dense sets. Successively choose closed balls
$$
\overline B(x_n,r_n)\subset B(x_{n-1},r_{n-1})\cap G_n,
\qquad 0<r_n<2^{-n},
$$
starting with a ball contained in $V\cap G_1$. The centres form a Cauchy sequence. Its limit belongs to every closed ball, and hence to $V\cap\bigcap_nG_n$.

= Closed convex absorbing set has an origin neighbourhood
{parent=Baire category theorem}

Let $S$ be a nonempty closed, convex, symmetric subset of a Banach space and suppose $\bigcup_{n\geq1}nS=X$. Baire's theorem puts a ball $B(x,r)$ inside some $nS$. Symmetry also puts $B(-x,r)$ there, and convexity puts the midpoint of $x+z$ and $-x+z$, namely $z$, in $nS$ whenever $\|z\|<r$. Hence $B(0,r/n)\subset S$.

= Closed symmetric absorbing set without an origin neighbourhood
{parent=Closed convex absorbing set has an origin neighbourhood}

Convexity is essential. In $\mathbb R$, let
$$
S=\{0\}\cup\bigcup_{k\geq0}
\{x:4^{-k}\leq|x|\leq2\cdot4^{-k}\}.
$$
This set is closed, symmetric, and has gaps arbitrarily near zero. Given $x\ne0$, choose $k$ so small-scale that the interval
$[|x|/(2\cdot4^{-k}),|x|/4^{-k}]$ has length at least one; it contains an integer $n$, and then $x/n\in S$. Thus $\bigcup_n nS=\mathbb R$, although $S$ is not a neighbourhood of zero.

= Isolated points in a countable complete metric space
{parent=Baire category theorem}

Every nonempty countable complete metric space has an <isolated point>. Otherwise it is the countable union of its singleton subsets, each of which is nowhere dense, contradicting the <Baire category theorem>.

If such a space has infinitely many points, it has infinitely many isolated points. Indeed, after deleting any finite collection of isolated points, the remaining set is closed, complete, countable, and nonempty. It therefore has a point isolated in the remaining space; because the deleted set is finite, that point is also isolated in the original space.

= Nowhere dense set
{parent=Baire category theorem}
{wiki}

A subset $A$ of a topological space is nowhere dense when the interior of its closure is empty. Equivalently, every nonempty open set contains a nonempty open subset disjoint from $A$.

= Meagre set
{parent=Baire category theorem}
{wiki}

A meagre, or first-category, set is a countable union of <nowhere dense sets>.

= Generic nowhere-monotone continuous function
{parent=Baire category theorem}

In $C([0,1])$ with the uniform norm, the continuous functions that are monotone on some interval of positive length form a <meagre set>. Indeed, it is enough to use intervals with rational endpoints. For each such interval, the nondecreasing and nonincreasing functions form closed sets with empty interior: a sufficiently small local triangular perturbation breaks the relevant inequality. The <Baire category theorem> therefore shows that a dense set of continuous functions is monotone on no nontrivial interval.

= Smooth function with a pointwise vanishing derivative
{parent=Baire category theorem}

Let $f\in C^\infty(\mathbb R)$ and suppose that for every $x$ there is an $n=n(x)\geq0$ such that $f^{(n)}(x)=0$. Then $f$ is a polynomial.

Set $S_n=\{x:f^{(n)}(x)=0\}$ and $E_n=\operatorname{int}S_n$. The sets $S_n$ are closed, $E_n\subset E_{n+1}$, and Baire's theorem on each interval shows that $\Omega=\bigcup_nE_n$ is dense. On every connected component of $\Omega$, compactness and the increasing cover by the $E_n$ show that one derivative vanishes locally with a uniform finite order, so $f$ agrees there with one polynomial.

If $F=\mathbb R\setminus\Omega$ were nonempty, it would be closed and have no isolated points: polynomial pieces on both sides of an isolated point would have matching derivatives of every order and hence join into one polynomial piece. Applying Baire's theorem to the cover $F=\bigcup_n(F\cap S_n)$ gives an interval $U$ and an index $k$ for which $F\cap U$ is nonempty and contained in $S_k$. Because $F$ has no isolated points, difference quotients show that every derivative of order at least $k$ vanishes on $F\cap U$. Each polynomial component of $\Omega$ meeting $U$ has an endpoint in $F\cap U$, so its degree is less than $k$. Thus $f^{(k)}=0$ throughout $U$, contradicting $F\cap U\ne\varnothing$. Hence $\Omega=\mathbb R$, and its sole connected component carries one polynomial.

= Space of smooth functions on a compact interval
{title2=$C^\infty([0,1])$}
{parent=Complete metric space}

The space $C^\infty([0,1])$ becomes a complete metric space under
$$
d(f,g)=\sum_{r=0}^\infty2^{-r}
\min\{1,\|f^{(r)}-g^{(r)}\|_\infty\}.
$$
Convergence in this metric is exactly uniform convergence of every derivative separately.

= Completeness of the smooth-function metric
{parent=Space of smooth functions on a compact interval}

If $(f_n)$ is Cauchy in the smooth-function metric, each derivative sequence $(f_n^{(r)})$ converges uniformly to a continuous function $g_r$. Passing to the limit in
$$
f_n^{(r)}(x)-f_n^{(r)}(0)=\int_0^x f_n^{(r+1)}(t)\,dt
$$
shows that $g_r'=g_{r+1}$. Hence $g_0$ is smooth and $f_n\to g_0$ in the metric.

= Generic superfactorial derivative growth at rational points
{parent=Space of smooth functions on a compact interval}

Outside a <meagre set> in $C^\infty([0,1])$, every rational $q\in(0,1)$ and every positive integer $M$ admit $m\geq M$ such that
$$
|f^{(m)}(q)|>m!m^m.
$$
For fixed $q,M$, the union over $m\geq M$ of these strict-inequality sets is open. It is dense because a perturbation $\delta\cos(K(x-q)-m\pi/2)$ can make the $m$th derivative arbitrarily large while keeping any prescribed finite collection of lower derivatives arbitrarily small. The <Baire category theorem> applied over the countable pairs $(q,M)$ gives the claim.

= Nowhere-analytic generic smooth function
{parent=Generic superfactorial derivative growth at rational points}

The generic derivative bound implies that at every rational $q\in(0,1)$,
$$
\limsup_m\left|\frac{f^{(m)}(q)}{m!}\right|^{1/m}=\infty.
$$
The <Cauchy-Hadamard theorem> therefore gives radius zero to the Taylor series at every such $q$. If a Taylor series represented $f$ on a neighborhood of any point, that neighborhood would contain a rational point at which $f$ was real analytic, a contradiction.

= Closed-subspace completeness theorem
{parent=Complete metric space}

A closed subspace of a complete metric space is complete. A Cauchy sequence in the subspace converges in the ambient space, and closedness keeps its limit in the subspace.

= Hausdorff distance
{c}
{parent=Metric space}
{wiki=Hausdorff_distance}

For nonempty bounded subsets of a metric space,
$$
d_H(A,B)=\max\left\{
\sup_{a\in A}d(a,B),
\sup_{b\in B}d(b,A)
\right\}.
$$
This is a metric on the nonempty closed bounded subsets. Without closedness it is only a pseudometric: a set and its closure have distance zero.

= Closed singleton embedding in a Hausdorff hyperspace
{parent=Hausdorff distance}

The map $x\mapsto\{x\}$ is an isometry into the hyperspace of nonempty closed bounded subsets. Its image is closed because a Hausdorff limit of singletons has diameter zero and is therefore a singleton.

= Completeness is reflected by the Hausdorff hyperspace
{parent=Closed singleton embedding in a Hausdorff hyperspace}

If the Hausdorff hyperspace is complete, its closed subspace of singletons is complete. Since that subspace is isometric to the original metric space, the original space is complete as well.

= Equicontinuous family of functions
{parent=Topological analysis}
{wiki=Equicontinuity}

A family $\mathcal F$ of maps between metric spaces is equicontinuous when, near each point, one input tolerance controls the output variation uniformly for every $f\in\mathcal F$. Uniform equicontinuity uses one input tolerance over the whole domain.

= Arzela-Ascoli theorem
{parent=Equicontinuous family of functions}
{c}
{wiki=Arzelà–Ascoli_theorem}

On a compact metric space, a uniformly bounded equicontinuous sequence of scalar continuous functions has a uniformly convergent subsequence. Equivalently, uniform boundedness and equicontinuity characterize relative compactness in $C(K)$.

= Diagonal subsequence for locally uniform convergence
{parent=Arzela-Ascoli theorem}

If successive subsequences converge uniformly on an exhaustion $K_1\subset K_2\subset\cdots$, the diagonal sequence converges uniformly on every fixed $K_m$, hence locally uniformly on their union.

= Escaping bump counterexample to global uniform convergence
{parent=Diagonal subsequence for locally uniform convergence}

For a compactly supported nonzero function $\psi$, the translates $f_n(x)=\psi(x-n)$ converge uniformly to zero on every compact subset of $\mathbb R$, while $\lVert f_n\rVert_\infty=\lVert\psi\rVert_\infty$ prevents uniform convergence on all of $\mathbb R$.

= Functional analysis
{parent=Analysis}
{wiki}

= Integral operator
{parent=Functional analysis}
{wiki=Integral_transform}

An integral operator maps a function $f$ to a function of the form
$$
(Tf)(x)=\int K(x,y)f(y)\,dy.
$$
A bounded continuous kernel on a compact domain defines a bounded operator in the uniform norm.

= Normed vector space
{parent=Functional analysis}
{wiki}

A normed vector space is a vector space equipped with a norm, whose induced metric is $d(x,y)=\lVert x-y\rVert$.

= L1 norm
{title2=$\lVert x\rVert_1$}
{parent=Normed vector space}
{c}
{wiki=Taxicab_geometry}

For $x\in\mathbb R^d$, the L1 norm is
$$
\lVert x\rVert_1=\sum_{j=1}^d|x_j|.
$$

= Euclidean norm
{title2=$\lVert x\rVert_2$}
{parent=Normed vector space}
{wiki=Euclidean_distance}

The Euclidean norm of $x=(x_1,\ldots,x_n)\in\mathbb R^n$ is
$$
\lVert x\rVert_2=\sqrt{x_1^2+\cdots+x_n^2}.
$$

= Euclidean normed vector space
{synonym}

= Euclidean ball
{parent=Euclidean norm}
{wiki=Ball_(mathematics)}

The closed Euclidean ball of radius $R$ about $a\in\mathbb R^n$ is $\{x:\lVert x-a\rVert_2\leq R\}$; replacing $\leq$ by $<$ gives the open ball.

= Supremum norm
{title2=$\lVert f\rVert_\infty$}
{parent=Normed vector space}
{wiki=Uniform_norm}

The supremum norm of a bounded real- or complex-valued <function> on a set $X$ is
$$
\lVert f\rVert_\infty=\sup_{x\in X}|f(x)|.
$$
For a <continuous function> on a <compact space>, the <extreme value theorem> makes this supremum a maximum.

= Uniform norm
{synonym}

= Surjective isometry of normed vector spaces
{parent=Normed vector space}

A surjective isometry $u:V\to W$ between normed vector spaces preserves every distance:
$$
\|u(v)-u(w)\|=\|v-w\|.
$$

= Mazur-Ulam theorem
{parent=Surjective isometry of normed vector spaces}
{c}
{wiki=Mazur–Ulam_theorem}

Every surjective isometry between real normed vector spaces is affine. In particular, if $u(0)=0$, then $u$ is real-linear.

= Metric extraction of a midpoint by shrinking diameters
{parent=Mazur-Ulam theorem}

For $v,w$ in a real normed space, start with the points at distance $\|v-w\|/2$ from both endpoints and repeatedly retain those within half the previous <diameter of a metric space>[diameter] of every retained point. Reflection about $(v+w)/2$ preserves every stage, the midpoint belongs to every stage, and the diameters decrease by a factor at least two. Their intersection is therefore exactly the midpoint.

= Nonsurjective isometry need not preserve midpoints
{parent=Surjective isometry of normed vector spaces}

Surjectivity in the <Mazur-Ulam theorem> is essential. The map
$$
u:\mathbb R\longrightarrow(\mathbb R^2,\|\cdot\|_\infty),
\qquad
u(x)=(x,|x|),
$$
preserves all distances and fixes zero, but it is not linear and does not preserve every midpoint.

= Banach space
{c}
{parent=Normed vector space}
{wiki}

A Banach space is a normed vector space that is complete in its norm metric.

= Banach spaces
{c}
{synonym}

= Uniform boundedness principle
{parent=Banach space}
{c}
{wiki}

If a family of bounded linear operators from a Banach space to a normed space is pointwise bounded, then their operator norms are uniformly bounded.

= Stone-Weierstrass theorem
{c}
{parent=Functional analysis}
{wiki=Stone%E2%80%93Weierstrass_theorem}

A point-separating real subalgebra of $C(X)$ that contains the constants is uniformly dense when $X$ is compact Hausdorff. Polynomial approximation of the square root makes its closure a lattice; finite maxima and minima then turn pointwise interpolation into uniform approximation.

= Weierstrass approximation theorem
{parent=Stone-Weierstrass theorem}
{c}
{wiki}

Every continuous real-valued function on a compact interval is a <uniform limit> of polynomials.

= Bernstein polynomial
{parent=Weierstrass approximation theorem}
{c}
{wiki=Bernstein_polynomial}

For $f\in C([0,1]^d)$, its tensor-product Bernstein polynomial is the expectation of $f(X_1/n,\ldots,X_d/n)$ for independent binomial variables $X_j\sim\operatorname{Bin}(n,x_j)$. Uniform continuity and concentration of the binomial variables prove uniform convergence to $f$.

= Open mapping theorem
{disambiguate=functional analysis}
{parent=Functional analysis}
{wiki}

A bounded surjective linear map between Banach spaces is open. Baire category first puts a ball in the closure of an image; iterative correction removes the closure.

= Open mapping theorem
{synonym}

= Bounded inverse theorem
{parent=open-mapping-theorem-functional-analysis}
{wiki}

A bounded bijective linear map between Banach spaces has a bounded inverse. This follows because an open bijection has continuous inverse.

= Diagonal operator on sequence space
{parent=Functional analysis}

A diagonal operator on $\ell^p$ acts coordinatewise as $(x_n)\mapsto(a_nx_n)$. It is bounded when $(a_n)$ is bounded, while its inverse on its image is bounded only when the nonzero $a_n$ are bounded away from zero.

= Unbounded inverse on a nonclosed operator range
{parent=Diagonal operator on sequence space}

The injective diagonal map $(x_n)\mapsto(x_n/n)$ on $\ell^2$ is bounded, but its inverse on the range is unbounded; the range is correspondingly not closed.

= Graph of a linear operator
{parent=Functional analysis}
{wiki=Closed_graph_theorem}

The graph of $T:X\to Y$ is the linear subspace $\Gamma(T)=\{(x,Tx):x\in X\}$ of $X\times Y$.

= Closed graph theorem
{parent=Graph of a linear operator}
{wiki}

A linear map between Banach spaces is continuous if and only if its graph is closed.

= Closed graph theorem from the bounded inverse theorem
{parent=Closed graph theorem}

For a closed graph, the projection $\Gamma(T)\to X$ is a bounded bijection between Banach spaces. Its bounded inverse followed by projection to $Y$ is $T$.

= Closed-graph proof for a factored operator
{parent=Closed graph theorem}

If bounded $S:X\to Y$ is injective and $\operatorname{im}T\subseteq\operatorname{im}S$, then $R=S^{-1}T$ has closed graph: $x_n\to x$ and $Rx_n\to z$ imply $Sz=Tx=S(Rx)$ and hence $z=Rx$.

= Incomplete-domain counterexample to closed-graph factorization
{parent=Closed graph theorem}

On incomplete $c_{00}$ with the $\ell^2$ norm, let $S(x_n)=(x_n/n)$ and let $T$ be inclusion into $\ell^2$. Their images agree, but $S^{-1}T(e_n)=ne_n$ is unbounded.

= Continuity of a positive linear map into a dual space
{parent=Closed graph theorem}

Let $V$ be a real Banach space and $T:V\to V^*$ be linear with $T_v(v)\geq0$ for every $v$. If $v_n\to0$ and $T_{v_n}\to f$ in $V^*$, positivity of $T_{v_n+tw}(v_n+tw)$ and passage to the limit give
$$
0\leq t f(w)+t^2T_w(w)
$$
for every real $t$ and every $w$. Taking small $t$ of either sign forces $f(w)=0$. Thus the graph of $T$ is closed, and the <closed graph theorem> makes $T$ continuous.

= Closed-range criterion for an injection
{parent=Functional analysis}

A bounded injective map between Banach spaces is bounded below exactly when its image is closed, by the bounded inverse theorem applied to its image.

= Neumann-series perturbation
{parent=Functional analysis}

If $A$ is invertible and $\|A^{-1}E\|<1$, then $A+E$ is invertible by the Neumann series. Similar iterative corrections prove stability of surjectivity.

= Density by annihilators
{parent=Functional analysis}

A linear subspace is dense exactly when the only continuous functional vanishing on it is zero, by Hahn-Banach separation.

= Hilbert space
{parent=Functional analysis}
{c}
{wiki}

A Hilbert space is a complete inner-product space.

= Closest point theorem in a Hilbert space
{parent=Hilbert space}
{wiki=Hilbert_projection_theorem}

Every nonempty closed convex subset $C$ of a Hilbert space contains a unique point nearest to each $x$. A minimizing sequence is Cauchy by the parallelogram identity, and uniqueness follows by applying the same identity to two minimizers and their midpoint.

= Orthogonal decomposition by a closed subspace
{parent=Closest point theorem in a Hilbert space}

For a closed subspace $F$ of a Hilbert space,
$$
H=F\oplus F^\perp.
$$
The closest point $P_Fx$ gives the first component, and differentiating $\|x-P_Fx-tu\|^2$ at zero shows that the remainder is orthogonal to every $u\in F$.

= Riesz representation theorem
{c}
{parent=Hilbert space}
{wiki}

Every bounded linear functional on a Hilbert space is inner product with one unique vector, with equal norms.

= Adjoint operator
{parent=Riesz representation theorem}
{wiki}

For $T\in L(H)$, the adjoint $T^*$ is the unique bounded operator satisfying
$$
\langle Tx,y\rangle=\langle x,T^*y\rangle.
$$
Riesz representation applied for each $y$ proves existence; in an orthonormal basis its matrix is the conjugate transpose of the matrix of $T$.

= Hermitian operator
{parent=Adjoint operator}
{c}
{wiki=Self-adjoint_operator}

A bounded operator is Hermitian, or self-adjoint, when $T=T^*$.

= Hermitian matrix
{synonym}

= Adjoint criterion for an invariant orthogonal complement
{parent=Adjoint operator}

For a subspace $U$ of a finite-dimensional inner-product space,
$$
T(U)\subseteq U
\quad\Longleftrightarrow\quad
T^*(U^\perp)\subseteq U^\perp.
$$
This follows directly by moving $T$ across the inner product and using $(U^\perp)^\perp=U$.

= Normal operator
{parent=Adjoint operator}
{wiki}

An operator is normal when $TT^*=T^*T$. Equivalently,
$$
\lVert Tx\rVert=\lVert T^*x\rVert
$$
for every $x$; the converse follows by polarizing the quadratic form of the self-adjoint commutator $T^*T-TT^*$.

= Spectral theorem for normal operators
{parent=Normal operator}
{wiki=Spectral_theorem}

Every normal operator on a finite-dimensional complex inner-product space has an orthonormal basis of eigenvectors. An eigenvector is also an eigenvector of the adjoint with conjugate eigenvalue, so its orthogonal complement reduces the operator and induction applies.

= Weak convergence in a Hilbert space
{parent=Hilbert space}
{wiki=Weak_convergence_(Hilbert_space)}

A sequence $x_n$ converges weakly to $x$ when $\langle x_n,y\rangle\to\langle x,y\rangle$ for every $y$ in the Hilbert space.

= Coordinate criterion for weak convergence in a separable Hilbert space
{parent=Weak convergence in a Hilbert space}

For a Hilbertian basis $(e_i)$, a sequence $x_n$ converges weakly if and only if $(x_n)$ is norm bounded and every coordinate sequence $\langle x_n,e_i\rangle$ converges.

= Weak subsequence of a bounded Hilbert-space sequence
{parent=Weak convergence in a Hilbert space}

Every bounded sequence in a separable Hilbert space has a weakly convergent subsequence. Successive subsequences make each basis coordinate converge, and a diagonal subsequence converges in every coordinate; the <coordinate criterion for weak convergence in a separable Hilbert space> finishes the proof.

= Weak lower semicontinuity of the Hilbert norm
{parent=Weak convergence in a Hilbert space}

If $x_n\rightharpoonup x$ in a Hilbert space, then
$$
\lVert x\rVert\leq\liminf_n\lVert x_n\rVert.
$$
For $x\ne0$, take the inner product with $x$ and apply Cauchy-Schwarz before passing to the lower limit.

= Radon-Riesz theorem
{parent=Weak convergence in a Hilbert space}
{c}
{wiki=Radon–Riesz_property}

In a Hilbert space, weak convergence $x_n\rightharpoonup x$ together with $\lVert x_n\rVert\to\lVert x\rVert$ implies strong convergence $x_n\to x$.

= Uniform basis-tail criterion for strong convergence
{parent=Radon-Riesz theorem}

Suppose $x_n\rightharpoonup x$ in a separable Hilbert space with Hilbertian basis $(e_i)$. Then $x_n\to x$ in norm exactly when
$$
\forall\epsilon>0\ \exists I\ 
\forall n,\qquad
\sum_{i\geq I}|\langle x_n,e_i\rangle|^2<\epsilon.
$$
The condition prevents norm from escaping to successively higher basis coordinates.

= Weakly null orthonormal sequence
{parent=Weak convergence in a Hilbert space}

Every <orthonormal sequence> in a Hilbert space converges weakly to zero, by <Bessel inequality>, but no subsequence converges strongly because distinct terms remain distance $\sqrt2$ apart.

= Mazur lemma
{c}
{parent=Weak convergence in a Hilbert space}
{wiki=Mazur%27s_lemma}

If $x_n$ converges weakly to $x$ in a Banach space, there are convex combinations of each tail $\{x_n,x_{n+1},\ldots\}$ that converge in norm to $x$.

= Norm-closed convex set is weakly closed
{parent=Weak convergence in a Hilbert space}

A norm-closed convex subset of a Banach space contains every weak limit of its sequences. Apply <Mazur lemma> to obtain norm-convergent convex combinations that remain in the set.

= Orthonormal sequence
{parent=Hilbert space}
{wiki=Orthonormality}

An orthonormal sequence $(e_n)$ satisfies $\langle e_n,e_m\rangle=0$ for $n\ne m$ and $\lVert e_n\rVert=1$.

= Hilbertian basis
{parent=Orthonormal sequence}
{wiki=Orthonormal_basis}

A Hilbertian basis is a complete orthonormal family: its closed linear span is the whole Hilbert space, equivalently every vector is the norm-convergent sum of its Fourier coefficients against the basis.

= Parseval identity for a Hilbertian basis
{parent=Hilbertian basis}
{c}

For a Hilbertian basis $(e_i)$,
$$
\langle x,y\rangle
=\sum_i\langle x,e_i\rangle\overline{\langle y,e_i\rangle},
\qquad
\lVert x\rVert^2=\sum_i|\langle x,e_i\rangle|^2.
$$

= Bessel inequality
{parent=Orthonormal sequence}
{c}
{wiki=Bessel%27s_inequality}

For an orthonormal sequence $(e_n)$ in a Hilbert space,
$$
\sum_n|\langle x,e_n\rangle|^2\leq\lVert x\rVert^2.
$$

= Compact operator
{parent=Functional analysis}
{wiki}

A bounded operator is compact when it maps the unit ball to a relatively compact set.

= Hilbert-Schmidt operator
{title2=$\lVert T\rVert_{\mathrm{HS}}$}
{parent=Compact operator}
{wiki=Hilbert%E2%80%93Schmidt_operator}

An operator on a Hilbert space is Hilbert--Schmidt when
$$
\lVert T\rVert_{\mathrm{HS}}^2=\sum_n\lVert Te_n\rVert^2<\infty
$$
for one, equivalently every, Hilbertian basis. Every Hilbert--Schmidt operator is compact.

= Finite-rank operator
{parent=Compact operator}
{wiki}

A finite-rank operator has finite-dimensional image and is compact.

= Spectral theorem for compact Hermitian operators
{parent=Compact operator}
{wiki=Spectral_theorem\#Compact_self-adjoint_operators}

A compact Hermitian operator has real nonzero eigenvalues of finite multiplicity, with zero as their only possible accumulation point, and an orthonormal eigenbasis after a basis of its kernel is included.

= Finite-rank truncation of a compact Hermitian operator
{parent=Spectral theorem for compact Hermitian operators}

If
$$
Tx=\sum_n\lambda_n\langle x,e_n\rangle e_n,
\qquad \lambda_n\to0,
$$
then truncating the sum after $N$ terms gives finite-rank Hermitian $T_N$ with
$$
\lVert T-T_N\rVert=\sup_{n>N}|\lambda_n|\to0.
$$

= Coordinate-projection approximation of a compact operator
{parent=Compact operator}

Let $P_N$ project $\ell^2$ onto its first $N$ coordinates. Although $P_N\to I$ only strongly on the whole unit ball, convergence is uniform on every compact subset. Hence compact $T$ satisfies
$$
\lVert P_NT-T\rVert\to0,
$$
and each $P_NT$ has finite rank.

= Sublinear functional
{parent=Functional analysis}
{wiki=Sublinear_function}

A real-valued functional $p$ is sublinear when
$p(x+y)\leq p(x)+p(y)$ and $p(tx)=tp(x)$ for every $t\geq0$.

= Hahn-Banach theorem
{c}
{parent=Functional analysis}
{wiki=Hahn–Banach_theorem}

Every bounded linear functional on a linear subspace of a real normed space extends to the whole space without increasing its norm.

= Canonical embedding into the bidual
{parent=Hahn-Banach theorem}

For a normed space $X$, the map $J:X\to X''$ defined by
$$
(Jx)(f)=f(x)
$$
is a linear isometry. The easy inequality is $\|Jx\|\leq\|x\|$; Hahn--Banach extends the norm-one functional on $\operatorname{span}\{x\}$ that takes $x$ to $\|x\|$, proving the reverse inequality.

= Singular functional on L infinity
{parent=Hahn-Banach theorem}

Point evaluation at zero is bounded on $C([0,1])$ with the essential-supremum norm, and Hahn--Banach extends it to a bounded functional on $L^\infty([0,1])$. No $g\in L^1$ represents this extension: continuous functions supported in shrinking neighbourhoods of zero have value one at zero while their integrals against $g$ tend to zero. Hence $(L^\infty)'$ is strictly larger than $L^1$.

= Hahn-Banach separation theorem
{c}
{parent=Hahn-Banach theorem}
{wiki=Hahn%E2%80%93Banach_theorem#Hahn%E2%80%93Banach_separation_theorem}

A point outside a nonempty closed convex subset of a normed space can be strictly separated from it by the real part of a bounded linear functional.

= Banach-Alaoglu theorem
{c}
{parent=Functional analysis}
{wiki=Banach–Alaoglu_theorem}

The closed unit ball of a dual Banach space is compact in the weak-star topology.

= Lax-Milgram theorem
{c}
{parent=Functional analysis}
{wiki=Lax–Milgram_theorem}

A bounded coercive bilinear form $a$ on a Hilbert space $H$ represents every bounded linear functional uniquely: for each $\ell\in H'$, there is a unique $u\in H$ with $a(u,v)=\ell(v)$ for every $v\in H$.

= Weak Dirichlet problem for the massive Laplacian
{parent=Lax-Milgram theorem}

For a bounded open $U\subset\mathbb R^n$, $m^2>0$, and $f\in L^2(U)$, there is a unique $u\in H_0^1(U)$ such that
$$
\int_U(\nabla u\cdot\nabla v+m^2uv)
=\int_Ufv
$$
for every $v\in H_0^1(U)$.

= Compact massive-Laplacian resolvent
{parent=Weak Dirichlet problem for the massive Laplacian}

The solution operator $(-\Delta+m^2)^{-1}:L^2(U)\to L^2(U)$ is compact on every bounded open set because it maps boundedly into $H_0^1(U)$ and the <Rellich-Kondrashov compactness theorem for H01> embeds that space compactly into $L^2(U)$.

= Sobolev space
{c}
{parent=Functional analysis}
{wiki}

For real $s$, the $L^2$-based Sobolev space $H^s(\mathbb R^n)$ consists of tempered distributions satisfying
$$
\lVert u\rVert_{H^s}^2
=\int_{\mathbb R^n}(1+|\xi|^2)^s|\widehat u(\xi)|^2\,d\xi<\infty.
$$

= Sobolev derivative estimate
{parent=Sobolev space}

For every multi-index $\alpha$ and real $s$,
$$
\|D^\alpha u\|_{H^{s-|\alpha|}}
\leq\|u\|_{H^s}.
$$
This follows from the Fourier multiplier $(i\xi)^\alpha$.

= Massive Laplacian isomorphism on Sobolev spaces
{parent=Sobolev space}

For every real $s$,
$$
-\Delta+1:H^{s+2}(\mathbb R^n)\to H^s(\mathbb R^n)
$$
is an isomorphism. Its inverse is the Fourier multiplier $(1+|\xi|^2)^{-1}$.

= Compactness from bounded support and an H2 bound
{parent=Sobolev space}

A sequence bounded in $H^2(\mathbb R^n)$ whose members are supported in one bounded set has a subsequence converging strongly in $H^1(\mathbb R^n)$. Apply Rellich compactness simultaneously to the functions and their first derivatives.

= Sobolev trace theorem
{c}
{parent=Sobolev space}
{wiki=Trace_operator}

For $s>1/2$, restriction to the coordinate hyperplane extends uniquely to a bounded linear map
$$
\gamma:H^s(\mathbb R^n)\longrightarrow H^{s-1/2}(\mathbb R^{n-1}).
$$
In Fourier variables, Cauchy--Schwarz bounds the integral over the normal frequency because $(1+t^2)^{-s}$ is integrable exactly when $s>1/2$.

= Failure of an Lp hyperplane trace
{parent=Sobolev trace theorem}

For $1\leq p<\infty$, no bounded map from $L^p(\mathbb R^n)$ to $L^p(\mathbb R^{n-1})$ can agree with restriction on continuous functions. Indeed,
$$
u_k(x',x_n)=\phi(x')\psi(kx_n),\qquad \psi(0)=1,
$$
has a fixed nonzero trace while $\lVert u_k\rVert_p=k^{-1/p}\lVert\phi\rVert_p\lVert\psi\rVert_p\to0$.

= Regularity gain for one plus an even power of the Laplacian
{parent=Sobolev space}

If $(\Delta^m+1)u=f$ distributionally on $\mathbb R^n$, $m$ is even, and $f\in H^r$, then
$$
\widehat u(\xi)=\frac{\widehat f(\xi)}{1+|\xi|^{2m}}
$$
and $u\in H^{r+2m}(\mathbb R^n)$.

= Sobolev embedding theorem
{parent=Sobolev space}
{wiki}

If $s>d/2$, then $H^s(\mathbb R^d)$ embeds continuously into bounded continuous functions.

= Rellich-Kondrashov compactness theorem for H01
{parent=Sobolev embedding theorem}
{c}
{wiki=Rellich%E2%80%93Kondrachov_theorem}

For every bounded open $\Omega\subset\mathbb R^d$, the inclusion $H_0^1(\Omega)\hookrightarrow L^2(\Omega)$ is compact.

= Zero extension of H01
{parent=Rellich-Kondrashov compactness theorem for H01}

By the definition of $H_0^1(\Omega)$ as the closure of compactly supported smooth functions, extension by zero maps it continuously into $H^1(\mathbb R^d)$ without creating a boundary distribution.

= High-frequency control by a Sobolev derivative
{parent=Rellich-Kondrashov compactness theorem for H01}

For $u\in H^1(\mathbb R^d)$, Plancherel gives
$$
\int_{|\xi|>R}|\widehat u(\xi)|^2\,d\xi
\leq R^{-2}\lVert\nabla u\rVert_2^2.
$$

= Fourier proof of Rellich compactness
{parent=Rellich-Kondrashov compactness theorem for H01}
{c}

Weak convergence on a bounded domain gives pointwise convergence of Fourier transforms and dominated convergence on bounded frequency balls. A uniform derivative bound controls the complementary high-frequency tails.

= Weak lower semicontinuity of a bounded-domain Schrodinger energy
{parent=Rellich-Kondrashov compactness theorem for H01}

On bounded $\Omega$ with $V\in L^\infty(\Omega)$, weak convergence in $H_0^1(\Omega)$ makes the Dirichlet term lower semicontinuous and, by Rellich compactness, makes the potential term $\int Vu^2$ continuous. Thus
$$
E(u)=\int_\Omega(|Du|^2+Vu^2)
$$
is weakly lower semicontinuous.

= Constrained ground-state minimizer on a bounded domain
{parent=Weak lower semicontinuity of a bounded-domain Schrodinger energy}

The infimum of $E(u)$ over $\lVert u\rVert_2=1$ is attained. A minimizing sequence is bounded in $H_0^1$ because $V$ is bounded below; weak compactness, Rellich strong $L^2$ convergence, and weak lower semicontinuity complete the direct-method argument.

= Local compactness plus uniform tail control
{parent=Rellich-Kondrashov compactness theorem for H01}

Strong convergence on every bounded region combines with a tail estimate uniform in the sequence to give global strong convergence.

= Strong convergence from weak convergence and tightness
{parent=Local compactness plus uniform tail control}

If local compactness makes every subsequential local limit agree with the global weak limit and the $L^2$ mass is uniformly tight, weak convergence upgrades to strong convergence.

= Noncompactness of a Sobolev embedding by translation
{parent=Sobolev embedding theorem}

Translations of one compactly supported bump have equal Sobolev norms. Widely separated translates have disjoint supports and remain a fixed positive distance apart in $L^2$, obstructing compactness on an unbounded domain.

= Loss of compactness at infinity
{parent=Noncompactness of a Sobolev embedding by translation}

A bounded sequence can fail to have a strongly convergent subsequence because its mass escapes to infinity even when its local regularity is uniformly controlled.

= Vanishing Dirichlet energy by dilation on the line
{parent=Noncompactness of a Sobolev embedding by translation}

For $\phi\in H^1(\mathbb R)$ with $\lVert\phi\rVert_2=1$, the dilation
$$
u_R(x)=R^{-1/2}\phi(x/R)
$$
preserves the $L^2$ norm and satisfies $\lVert u_R'\rVert_2^2=R^{-2}\lVert\phi'\rVert_2^2$. Hence normalized functions on the line have Dirichlet-energy infimum zero, which no nonzero $L^2$ function attains.

= Hölder space
{title2=$C^{0,\alpha}$}
{parent=Sobolev embedding theorem}
{c}
{wiki=H%C3%B6lder_condition}

A bounded function belongs to $C^{0,\alpha}(\mathbb R^n)$ when
$$
\sup_{x\ne y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<\infty.
$$

= Fourier proof of Hölder regularity from a Sobolev norm
{parent=Hölder space}
{c}

If $s>n/2+\alpha$ with $0<\alpha\leq1$, then Fourier inversion, $|e^{it}-1|\leq C_\alpha|t|^\alpha$, and Cauchy-Schwarz give
$$
\lVert u\rVert_{C^{0,\alpha}}
\leq C\lVert u\rVert_{H^s}.
$$
The weighted frequency integral converges precisely because $s-\alpha>n/2$.

= Ladyzhenskaya inequality in two dimensions
{c}
{parent=Sobolev space}
{wiki=Ladyzhenskaya%27s_inequality}

For $u\in C_c^\infty(\mathbb R^2)$,
$$
\lVert u\rVert_4^4
\leq4\lVert u\rVert_2^2\lVert\nabla u\rVert_2^2.
$$
Apply the fundamental theorem of calculus to $|u|^2$ in each coordinate, multiply the resulting one-dimensional bounds, integrate, and use Cauchy--Schwarz.

= Space of continuous functions vanishing at infinity
{parent=Functional analysis}
{wiki}

C0(X) consists of continuous functions whose values become arbitrarily small outside compact sets.

= Completeness of continuous functions vanishing at infinity
{parent=Space of continuous functions vanishing at infinity}

With the supremum norm, $C_0(\mathbb R^d)$ is complete. A uniform Cauchy sequence has a uniform continuous limit, and one member controls the limit uniformly outside a large ball. Every member is also uniformly continuous: use uniform continuity on a large compact ball and smallness of the function beyond it.

= Uniform square-root perturbation under linear growth
{parent=Space of continuous functions vanishing at infinity}

If $\varepsilon:\mathbb R\to[0,\infty)$ satisfies $\varepsilon(t)\leq M|t|$, then
$$
0\leq\sqrt{x^2+\varepsilon(x/n)}-|x|
=\frac{\varepsilon(x/n)}{\sqrt{x^2+\varepsilon(x/n)}+|x|}
\leq\frac Mn.
$$
Thus these square-root perturbations converge uniformly to $|x|$.

= Noncompactness by separated translates
{parent=Functional analysis}
{wiki}

Translations of a compactly supported bump can form an infinite family separated by a fixed sup-norm distance, disproving compactness.

= Continuous dual space
{parent=Functional analysis}
{wiki=Continuous_dual_space}

The continuous dual $X^*$ is the normed space of bounded linear scalar-valued functionals on $X$.

= Positive linear functional
{parent=Continuous dual space}
{wiki=Positive_linear_functional}

A linear functional $\Lambda$ on an ordered vector space of functions is positive when $f\geq0$ implies $\Lambda(f)\geq0$. Positivity implies monotonicity: $f\leq g$ gives $\Lambda(f)\leq\Lambda(g)$.

= Operator norm
{parent=Continuous dual space}
{wiki}

The operator norm is $\lVert T\rVert=\sup_{\lVert x\rVert\leq1}\lVert Tx\rVert$.

= Matrix 2-norm
{title2=$\|A\|_2$}
{parent=Operator norm}
{wiki=Matrix_norm}

The matrix 2-norm is the <operator norm> induced by the <Euclidean norm>:
$$
\|A\|_2=\sup_{x\ne0}\frac{\|Ax\|_2}{\|x\|_2}.
$$
It equals the largest <singular value> of $A$ and is invariant under multiplication by <orthogonal matrix>[orthogonal] or <unitary operator>[unitary] matrices.

= Completeness of the dual space
{parent=Continuous dual space}

The continuous dual of every normed space is Banach because an operator-norm Cauchy sequence converges pointwise to a bounded linear functional and then uniformly on the unit ball.

= Duality of sequence spaces
{parent=Continuous dual space}
{wiki=Lp_space\#Dual_spaces}

Coordinate pairing gives $(\ell^p)^*=\ell^q$ for $1<p<\infty$, $(\ell^1)^*=\ell^\infty$, and $(c_0)^*=\ell^1$.

= Duality of Lp spaces
{title2=$(L^p)^*=L^q$}
{parent=Continuous dual space}
{wiki=Lp_space#Dual_spaces}

For $1<p<\infty$ and conjugate exponent $q$, every bounded linear functional on $L^p$ has the form
$$
f\longmapsto\int f\omega
$$
for a unique $\omega\in L^q$, and its operator norm is $\|\omega\|_q$.

= Positive functional representation on Lp
{parent=Duality of Lp spaces}

On a finite measure space, a positive functional $\Lambda\in(L^p)^*$ defines the finite measure $\nu(E)=\Lambda(\mathbf1_E)$. The <Radon-Nikodym theorem> gives $d\nu=\omega\,d\mu$ with $\omega\geq0$. The bound
$$
\int|f|\omega=\Lambda(|f|)
\leq\|\Lambda\|\,\|f\|_p
$$
and $L^p$-$L^q$ norm duality imply $\omega\in L^q$, after which density extends the integral representation to every $f\in L^p$.

= Holder inequality
{title2=$1/p+1/q=1$}
{parent=Functional analysis}
{c}
{wiki=Hölder%27s_inequality}

For conjugate exponents $p,q$,
$$
\int|fg|\leq\|f\|_p\|g\|_q,
$$
with the analogous inequality $\sum_n|x_ny_n|\leq\lVert x\rVert_p\lVert y\rVert_q$ for sequences.

= Conjugate exponents
{title2=$1/p+1/q=1$}
{parent=Holder inequality}

Exponents $p,q\in[1,\infty]$ are conjugate when $1/p+1/q=1$, with the convention $1/\infty=0$.

= Norming vector for Holder inequality
{parent=Holder inequality}

For nonzero $y\in\ell^q$, a normalized sequence proportional to $\operatorname{sgn}(y_n)|y_n|^{q-1}$ attains equality in Hölder's inequality.

= Minkowski integral inequality
{parent=Holder inequality}
{c}
{wiki=Minkowski_inequality}

For suitable measurable $F(x,y)$ and $1\leq p<\infty$,
$$
\left\|\int F(x,\mathord\cdot)\,dx\right\|_p
\leq\int\|F(x,\mathord\cdot)\|_p\,dx.
$$
Pair the left side with a unit vector in the dual $L^q$ space, use Tonelli's theorem, and apply Hölder's inequality in $y$.

= Space of sequences converging to zero
{parent=Functional analysis}
{wiki=C0_space\#Sequences}

The Banach space $c_0$ consists of scalar sequences tending to zero with the supremum norm.

= Convergent sequence space
{parent=Space of sequences converging to zero}
{wiki=Sequence_space}

The space $c$ consists of convergent scalar sequences with the supremum norm.

= Absorption of a finite-dimensional summand by c0
{parent=Convergent sequence space}

The map $(x_n)\mapsto(\lim x_n,x_1-\lim x_n,x_2-\lim x_n,\ldots)$ proves $c\cong c_0$, equivalently $c_0\oplus\mathbb F\cong c_0$.

= Banach space isomorphism
{parent=Functional analysis}
{c}
{wiki}

A Banach space isomorphism is a bounded linear bijection with bounded inverse.

= Fourier analysis
{parent=Analysis}
{c}
{wiki}

= Fourier mode
{parent=Fourier analysis}
{wiki=Fourier_series}

A spatial Fourier mode has the form $e^{ikx}$ and is an eigenfunction of every constant-coefficient spatial differential operator; in particular, $\partial_x^2e^{ikx}=-k^2e^{ikx}$.

= Schwartz space
{title2=$\mathcal S(\mathbb R^n)$}
{c}
{parent=Fourier analysis}
{wiki}

The Schwartz space $\mathcal S(\mathbb R^n)$ consists of smooth functions whose derivatives decay faster than every inverse polynomial. It is dense in every $H^s(\mathbb R^n)$.

= Tempered distribution
{title2=$\mathcal S'(\mathbb R^n)$}
{parent=Schwartz space}
{wiki}

A tempered distribution is a continuous linear functional on the <Schwartz space>. Tempered distributions include functions of at most polynomial growth and admit a <Fourier transform> by duality.

= Positive distribution
{parent=Tempered distribution}
{wiki=Distribution_(mathematics)}

A real distribution $u$ is positive when $u[\phi]\geq0$ for every nonnegative test function $\phi$. Every positive distribution has order zero and is represented locally by a positive measure.

= Fourier inversion theorem
{parent=Fourier analysis}
{c}
{wiki}

For the convention $\widehat f(\xi)=\int_{\mathbb R^n} e^{-ix\cdot\xi}f(x)dx$, if $f,\widehat f\in L^1$, then
$$
f(x)=\frac1{(2\pi)^n}\int_{\mathbb R^n}e^{ix\cdot\xi}\widehat f(\xi)d\xi
$$
almost everywhere. Reversing both exponential signs gives the equivalent opposite convention.

= Fourier inversion
{synonym}

= Continuous version of Fourier inversion
{parent=Fourier inversion theorem}

When $\widehat f\in L^1$, its inverse Fourier integral is continuous by dominated convergence. If $f$ is also continuous and Fourier inversion identifies the two functions almost everywhere, then they agree everywhere.

= Fourier transform of a derivative
{parent=Fourier inversion theorem}
{c}

For the angular-frequency convention and sufficient decay,
$$
\widehat{f'}(k)=ik\widehat f(k).
$$

= Inverse Fourier transforms of phase factors
{parent=Fourier inversion theorem}
{c}

For the angular-frequency convention,
$$
\mathcal F^{-1}[e^{ika}]=\delta(x+a),
\qquad
\mathcal F^{-1}[e^{-ika}]=\delta(x-a).
$$
Consequently cosine transforms to the half-sum of the two shifted deltas, while sine transforms to their signed difference divided by $2i$.

= Translation property of the Fourier transform
{parent=Fourier inversion theorem}
{c}

For the angular-frequency convention,
$$
\mathcal F[f(x-a)](k)=e^{-ika}\widehat f(k),
$$
so multiplication by a phase factor in frequency translates a function in physical space.

= Fourier transform of a triangular function
{parent=Fourier analysis}
{wiki=Triangular_function}

For $\theta_n(x)=(1-|x|/n)_+$,
$$
\widehat\theta_n(\xi)
=\frac{2(1-\cos(n\xi))}{n\xi^2}
=n\left(\frac{\sin(n\xi/2)}{n\xi/2}\right)^2,
$$
with value $n$ at zero.

= Triangular function
{title2=$\operatorname{tri}$}
{parent=Fourier transform of a triangular function}
{wiki}

The triangular function
$$
\operatorname{tri}(x)=(1-|x|)\mathbf1_{[-1,1]}(x)
$$
has angular-frequency Fourier transform $4\sin^2(\xi/2)/\xi^2$.

= Squared sinc function
{parent=Fourier transform of a triangular function}
{wiki=Sinc_function}

The squared sinc function is nonnegative and integrable; it occurs as the Fourier transform of a triangular function.

= Triangular frequency cutoff
{parent=Fourier transform of a triangular function}

The cutoffs $\theta_n(\xi)=(1-|\xi|/n)_+$ increase pointwise to one, while their Fourier transforms are nonnegative and have integral $2\pi$ under the angular-frequency convention.

= Positive-Fourier-transform L1 bound
{parent=Triangular frequency cutoff}

If $f\in L^1\cap L^\infty$ and $\widehat f\geq0$, testing against triangular frequency cutoffs and using monotone convergence gives $\lVert\widehat f\rVert_1\leq2\pi\lVert f\rVert_\infty$.

= Convolution
{parent=Fourier analysis}
{wiki}

The convolution of integrable functions on the real line is
$$
(f*g)(x)=\int_{-\infty}^{\infty}f(y)g(x-y)\,dy.
$$

= Approximate identity
{parent=Convolution}
{wiki}

An approximate identity is a family of integrable kernels $K_\varepsilon$ whose total mass is one and whose mass concentrates near zero as $\varepsilon\to0$. Under standard hypotheses, $f*K_\varepsilon$ converges to $f$ in norm and at almost every Lebesgue point.

= Discrete convolution
{parent=Convolution}
{wiki=Convolution#Discrete_convolution}

For sequences $a,b$, their discrete convolution is
$$
(a*b)_n=\sum_m a_{n-m}b_m.
$$

= Convolution theorem
{parent=Convolution}
{wiki}

For $\widehat f(\xi)=\int f(x)e^{-2\pi ix\cdot\xi}\,dx$, Fubini gives $\widehat{f*g}=\widehat f\,\widehat g$.

= Gamma density from repeated exponential convolution
{parent=Convolution theorem}
{wiki=Gamma_distribution}

For $F(x)=e^{-x}\mathbf 1_{x\geq0}$, induction in the convolution integral gives
$$
F^{*n}(x)=\frac{x^{n-1}e^{-x}}{(n-1)!}\mathbf 1_{x\geq0}.
$$
Its Fourier transform in the angular-frequency convention is $(1+ik)^{-n}$.

= Riemann-Lebesgue lemma
{c}
{parent=Fourier analysis}
{wiki=Riemann%E2%80%93Lebesgue_lemma}

If $f\in L^1(\mathbb R^n)$, then its Fourier transform is continuous and tends to zero as $|\xi|\to\infty$. Approximate $f$ in $L^1$ by a compactly supported smooth function; integration by parts makes the approximant's transform decay, while $\|\widehat f-\widehat g\|_\infty\leq\|f-g\|_1$ transfers the conclusion.

= Radial power integrability criterion
{parent=Riemann-Lebesgue lemma}

If a radial function on $\mathbb R^n$ behaves like $r^a$ near zero and like $r^{-b}$ at infinity, then its $q$th power is locally integrable at zero exactly when $aq>-n$ and integrable at infinity exactly when $bq>n$. This follows from the radial measure factor $r^{n-1}\,dr$.

= Interpolation between L2 and Linfinity by a pointwise bound
{title2=$L^2\cap L^\infty\subseteq L^p$}
{parent=Riemann-Lebesgue lemma}

For $2\leq p<\infty$,
$$
\|h\|_p^p
\leq\|h\|_\infty^{p-2}\|h\|_2^2.
$$
Thus every function in $L^2\cap L^\infty$ belongs to every $L^p$ between them.

= Plancherel theorem
{title2=$\|\widehat f\|_2=(2\pi)^{n/2}\|f\|_2$}
{parent=Fourier analysis}
{c}
{wiki}

For the unnormalized angular-frequency transform,
$$
\int_{\mathbb R^n}|\widehat f(\xi)|^2\,d\xi
=(2\pi)^n\int_{\mathbb R^n}|f(x)|^2\,dx.
$$
After normalization, the Fourier transform extends unitarily to $L^2$. For integrable $f$ and $\widehat f$, inversion and Fubini give the identity directly.

= Parseval identity
{parent=Fourier analysis}
{c}
{wiki}

For $f,g\in L^2(\mathbb R^n)$ and the convention $\widehat f(\xi)=\int f(x)e^{-ix\cdot\xi}\,dx$,
$$
\int_{\mathbb R^n}f(x)\overline{g(x)}\,dx
=\frac1{(2\pi)^n}\int_{\mathbb R^n}\widehat f(\xi)\overline{\widehat g(\xi)}\,d\xi.
$$

= Even rational Parseval integral
{parent=Parseval identity}
{c}

Applying the Parseval identity to the repeated exponential convolution gives
$$
\int_{-\infty}^{\infty}\frac{dk}{(1+k^2)^{n+1}}
=\frac{\pi(2n)!}{2^{2n}(n!)^2}
=\frac\pi{4^n}\binom{2n}{n}.
$$

= Fourier transform of a Gaussian
{parent=Fourier analysis}
{c}
{wiki=Gaussian_function\#Fourier_transform}

With $\widehat f(k)=\int_{\mathbb R}f(x)e^{-ikx}dx$ and $a>0$,
$$
\widehat{e^{-ax^2+ibx}}(k)
=\sqrt{\frac\pi a}\exp\left(-\frac{(k-b)^2}{4a}\right).
$$

= Complex analysis
{parent=Analysis}
{wiki}

= Polylogarithm
{title2=$\operatorname{Li}_s(z)$}
{parent=Complex analysis}
{wiki}

For $|z|<1$, the polylogarithm is $\operatorname{Li}_s(z)=\sum_{n\geq1}z^n/n^s$; contour formulas analytically continue it to a slit plane.

= Holomorphic function
{parent=Complex analysis}
{wiki}

A complex-valued <function> is holomorphic on an <open set> when it has a complex <derivative> at every point of that set.

= Analytic function
{synonym}

= Order of a zero of a holomorphic function
{parent=Holomorphic function}
{wiki=Zero_of_a_function#Multiple_zeros}

If a nonzero holomorphic function has expansion
$$
f(z)=(z-z_0)^m g(z),
\qquad g(z_0)\ne0,
$$
then $m$ is the order or multiplicity of its zero at $z_0$.

= Simple zero
{parent=Order of a zero of a holomorphic function}
{wiki=Zero_of_a_function#Simple_zeros}

A simple zero has order one, equivalently $f(z_0)=0$ and $f'(z_0)\ne0$.

= Analytic continuation
{parent=Holomorphic function}
{wiki}

An analytic continuation extends a <holomorphic function> through overlapping connected open sets while preserving its values on their overlap. Continuation along different paths can produce different germs when the domain contains <branch points>.

= Monodromy theorem
{parent=Analytic continuation}
{c}
{wiki}

If a function element can be analytically continued along every path in a domain, continuation along two endpoint-fixed homotopic paths gives the same terminal germ.

= Analytic continuation by contour deformation
{parent=Analytic continuation}

If a parameter-dependent contour integral has a moving pole, deforming the integration contour continuously so that the pole never crosses it preserves a holomorphic branch. Returning the contour to its original path after crossing a pole adds or subtracts the corresponding residue contribution.

= Multivalued inverse hyperbolic sine
{title2=$\operatorname{arcsinh}z$}
{parent=Analytic continuation}
{wiki=Inverse_hyperbolic_functions}

The solutions of $\sinh w=z$ are
$$
w=\operatorname{Arcsinh}z+2\pi ik
\quad\hbox{or}\quad
w=-\operatorname{Arcsinh}z+(2k+1)\pi i,
\qquad k\in\mathbb Z,
$$
for any chosen local branch $\operatorname{Arcsinh}$.

= Entire function
{parent=Holomorphic function}
{wiki}

An entire function is a <holomorphic function> whose domain is the whole <complex plane>.

= Little Picard theorem
{c}
{parent=Entire function}
{wiki=Picard_theorem}

Every nonconstant entire function takes every complex value with at most one exception.

= Great Picard theorem
{c}
{parent=Little Picard theorem}
{wiki=Picard_theorem}

Near an essential singularity, a holomorphic function takes every complex value, with at most one exception, infinitely often.

= Analytic function with image in an affine real line
{parent=Holomorphic function}

If $f=u+iv$ is a <holomorphic function> on a connected domain and
$$
au+bv=c,\qquad a^2+b^2\ne0,
$$
then the <Cauchy-Riemann equations> force both components of $f'$ to vanish. Hence $f$ is constant.

= Simply connected domain
{parent=Complex analysis}
{wiki=Simply_connected_space}

A domain is simply connected when every closed path in it can be continuously contracted to a point while remaining in the domain.

= Primitive of a holomorphic function on a simply connected domain
{parent=Simply connected domain}

Every <holomorphic function> $f$ on a <simply connected domain> has a single-valued <antiderivative>. Fixing $z_0$ and setting
$$
F(z)=\int_{z_0}^z f(t)\,dt
$$
gives a path-independent function with $F'=f$, because the <Cauchy integral theorem> makes the integral around every closed path zero.

= Complex inverse sine
{title2=$\arcsin z$}
{parent=Complex analysis}
{wiki=Inverse_trigonometric_functions}

The principal complex inverse sine on
$$
\mathbb C\setminus\bigl((-\infty,-1]\cup[1,\infty)\bigr)
$$
is the branch
$$
\arcsin z=\int_0^z\frac{dt}{\sqrt{1-t^2}}
$$
whose derivative equals one at zero.

= Branches of the complex inverse sine
{parent=Complex inverse sine}

If $G$ is the principal <complex inverse sine>, all values reached by <analytic continuation> are
$$
2n\pi+G(z)
\quad\hbox{or}\quad
(2n+1)\pi-G(z),
\qquad n\in\mathbb Z.
$$
This follows from the <monodromy reflection at a square-root branch point>[monodromy reflections] about the branch values $-\pi/2$ and $\pi/2$.

= Morera's theorem
{parent=Complex analysis}
{c}
{wiki=Morera%27s_theorem}

If a continuous complex-valued function on a domain has zero integral around the boundary of every triangle contained in that domain, then it is holomorphic.

= Morera theorem
{synonym}

= Cauchy integral theorem
{parent=Complex analysis}
{c}
{wiki=Cauchy%27s_integral_theorem}

If a function is holomorphic on a simply connected domain, its integral around every closed piecewise smooth contour in that domain vanishes. In particular, contours with common endpoints may be deformed through the domain without changing the integral.

= Gaussian contour translation
{parent=Cauchy integral theorem}
{c}

For $a>0$ and real $c$, integrate $e^{-az^2}$ around a wide rectangle between the real axis and $\operatorname{Im}z=c$. The vertical contributions vanish as the width tends to infinity, so
$$
\int_{\mathbb R+ic}e^{-az^2}\,dz
=\int_{\mathbb R}e^{-ax^2}\,dx.
$$

= Cauchy principal value
{c}
{parent=Complex analysis}
{wiki}

The principal value uses symmetric truncation at infinity and symmetric deletion around real singularities.

= Principal-value beta integral
{parent=Cauchy principal value}

For $0<s<a$,
$$\operatorname{PV}\int_0^\infty\frac{x^{s-1}}{1-x^a}dx=\frac\pi a\cot\frac{\pi s}{a}.$$

= Principal-value residue rule
{parent=Complex analysis}

For real $a$ and nonzero real $\omega$,
$$
\operatorname{PV}\int_{-\infty}^{\infty}\frac{e^{i\omega x}}{x-a}\,dx
=i\pi\operatorname{sgn}(\omega)e^{i\omega a}.
$$

= Gamma function
{title2=$\Gamma$}
{parent=Complex analysis}
{wiki}

$\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}\,dt$ for $\Re z>0$, and $\Gamma(z+1)=z\Gamma(z)$ gives meromorphic continuation.

= Gamma function recurrence
{c}
{parent=Gamma function}

Integration by parts gives
$$
\Gamma(z+1)=z\Gamma(z).
$$

= Bose integral
{c}
{parent=Gamma function}
{wiki=Polylogarithm#Integral_representation}

For $\operatorname{Re}s>1$, expansion of $(e^x-1)^{-1}$ as a <geometric series> and termwise integration give
$$
\int_0^\infty\frac{x^{s-1}}{e^x-1}\,dx
=\Gamma(s)\zeta(s).
$$

= Euler--Mascheroni constant
{parent=Gamma function}
{c}
{wiki=Euler%27s_constant}

The Euler--Mascheroni constant is
$$
\gamma=\lim_{n\to\infty}\left(\sum_{k=1}^n\frac1k-\log n\right).
$$

= Weierstrass product for the reciprocal gamma function
{parent=Gamma function}
{c}
{wiki=Weierstrass_product}

The reciprocal gamma function has the entire-product representation
$$
\frac1{\Gamma(z)}
=ze^{\gamma z}\prod_{k=1}^{\infty}
\left(1+\frac zk\right)e^{-z/k}.
$$

= Digamma function
{parent=Gamma function}
{wiki}

The digamma function is the logarithmic derivative
$$
\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}.
$$
Logarithmically differentiating the Weierstrass product gives
$$
\psi(z)=-\gamma-\frac1z
+z\sum_{k=1}^{\infty}\frac1{k(z+k)}.
$$

= Trigamma function
{parent=Digamma function}
{wiki}

The trigamma function is $\psi'(z)$. For real $z>0$,
$$
\psi'(z)=\frac1{z^2}+\sum_{k=1}^{\infty}\frac1{(z+k)^2}>0.
$$

= Positive zero of the digamma function
{parent=Digamma function}

The digamma function is strictly increasing on the positive real axis, while
$$
\psi(1)=-\gamma<0,
\qquad
\psi(2)=1-\gamma>0.
$$
It therefore has exactly one positive zero, and that zero lies in $(1,2)$.

= Gamma reflection formula
{parent=Gamma function}

$$
\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin\pi z}.
$$

= Gamma duplication formula
{parent=Gamma function}

$$
\Gamma(z)\Gamma(z+\tfrac12)=2^{1-2z}\sqrt\pi\,\Gamma(2z).
$$
It follows by making the logarithm of the quotient entire and periodic; growth forces its periodic remainder to be constant.

= Beta function
{title2=$B$}
{parent=Gamma function}
{wiki=Beta_function}

For $\operatorname{Re}p,\operatorname{Re}q>0$, Euler's beta function is
$$B(p,q)=\int_0^1t^{q-1}(1-t)^{p-1}\,dt.$$

= Beta--gamma identity
{parent=Beta function}

The sum-and-ratio change of variables in a product of gamma integrals gives
$$B(p,q)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}.$$

= Sum-and-ratio substitution for gamma integrals
{parent=Beta--gamma identity}

For $s,t>0$, put $r=s+t$ and $u=t/(s+t)$. Then $(s,t)=(r(1-u),ru)$ and $ds\,dt=r\,dr\,du$, separating radial gamma and ratio beta integrals.

= Logarithmic moments of the Cauchy kernel
{parent=Beta function}

For $0<a<2$,
$$
I(a)=\int_0^\infty\frac{x^{a-1}}{1+x^2}\,dx
=\frac\pi2\csc\frac{\pi a}{2}.
$$
Differentiating at $a=1$ and expanding the secant gives
$$
\int_0^\infty\frac{(\log x)^2}{1+x^2}\,dx=\frac{\pi^3}{8},
\qquad
\int_0^\infty\frac{(\log x)^4}{1+x^2}\,dx=\frac{5\pi^5}{32}.
$$

= Gamma function asymptotic at zero
{parent=Gamma function}

The functional equation and $\Gamma(1)=1$ give $\Gamma(z)\sim1/z$ as $z\to0$ away from the negative real axis.

= Diagonal beta-function asymptotic at zero
{parent=Gamma function asymptotic at zero}

The beta--gamma identity gives $B(z,z)=\Gamma(z)^2/\Gamma(2z)\sim2/z$ as $z\to0$ through the right half-plane.

= Riemann surfaces
{c}
{parent=Complex analysis}
{wiki=Riemann_surface}

= Holomorphic map
{parent=Riemann surfaces}
{wiki=Holomorphic_function#Maps_between_Riemann_surfaces}

A map between Riemann surfaces is holomorphic when its expression in every pair of local complex coordinates is a <holomorphic function>.

= Analytic map
{synonym}

= Germ of a holomorphic function
{parent=Riemann surfaces}
{wiki=Germ_(mathematics)}

A germ at $z\in D$ is an equivalence class $[f]_z$ of holomorphic functions defined near $z$, where two representatives are equivalent when they agree on some neighbourhood of $z$.

= Space of germs of holomorphic functions
{parent=Germ of a holomorphic function}

The space $\mathcal G$ of germs over a domain $D$ has basic open sheets
$$
\{[f]_z:z\in U\}
$$
for holomorphic $f$ on open $U\subseteq D$. The projection $\pi([f]_z)=z$ restricts to a homeomorphism on each sheet, and these projections form holomorphic coordinate charts.

= Evaluation map on a space of germs
{parent=Space of germs of holomorphic functions}

The evaluation map $\mathcal E([f]_z)=f(z)$ is holomorphic. On the sheet associated with $(U,f)$, its coordinate expression is exactly the holomorphic function $f$.

= Germ surface of the square root of z to the eighth minus one
{parent=Space of germs of holomorphic functions}

Over $D=\mathbb C\setminus\{z:z^8=1\}$, the two-valued square root is represented by
$$
R=\{(z,w)\in D\times\mathbb C:w^2=z^8-1\}.
$$
The maps $\pi(z,w)=z$ and $\mathcal E(z,w)=w$ identify this unbranched double cover analytically with the corresponding component of the space of germs.

= Complex structure lifted through a covering map
{parent=Riemann surfaces}

If $\pi:S\to R$ is a covering of a Riemann surface, compose every chart on an evenly covered open set with each local inverse sheet of $\pi$. The resulting transition maps are those of $R$, so they define a unique complex structure making $\pi$ a local biholomorphism.

= Uniformization theorem
{parent=Riemann surfaces}
{wiki}

Every simply connected Riemann surface is biholomorphic to the Riemann sphere, the complex plane, or the unit disc.

= Riemann mapping theorem
{c}
{parent=Uniformization theorem}
{wiki}

Every nonempty simply connected proper open subset of the <complex plane> is <conformal equivalence>[conformally equivalent] to the <unit disc>.

= Automorphisms of simply connected Riemann surfaces
{parent=Uniformization theorem}

The sphere has the Möbius group $PGL_2(\mathbb C)$, the plane has the affine maps $z\mapsto az+b$ with $a\ne0$, and the disc has the maps $e^{i\theta}(z-a)/(1-\bar az)$ with $|a|<1$.

= Riemann surfaces uniformized by the complex plane
{parent=Uniformization theorem}

The quotients of $\mathbb C$ by free properly discontinuous conformal actions are $\mathbb C$, $\mathbb C^*$, and the complex tori $\mathbb C/\Lambda$. The translation group has respectively rank zero, one, or two.

= Plane domain with two omitted points is hyperbolic
{parent=Uniformization theorem}

Every plane domain whose complement contains at least two points is uniformized by the unit disc. A plane universal cover would give a nonconstant entire function omitting two values, contrary to the <Little Picard theorem>.

= Compact Riemann surface containing an embedded punctured plane
{parent=Uniformization theorem}

If $\mathbb C^*$ embeds holomorphically in a compact Riemann surface $R$, the embedding extends across zero and infinity to a degree-one holomorphic map from the <Riemann sphere> to $R$. Hence $R$ is conformally the sphere.

= Identity theorem on a Riemann surface
{parent=Riemann surfaces}

Two holomorphic functions on a connected Riemann surface that agree on a set with an accumulation point agree everywhere. Local charts reduce the proof to isolated zeros of a one-variable holomorphic function.

= Harmonic function on a Riemann surface
{parent=Riemann surfaces}

A real-valued function on a Riemann surface is harmonic when its expression in every holomorphic coordinate chart has vanishing planar Laplacian.

= Conformal invariance of harmonicity
{parent=Harmonic function on a Riemann surface}

Under a holomorphic coordinate change $w=w(z)$,
$$
\Delta_z(H\circ w)=|w'(z)|^2(\Delta_wH)\circ w,
$$
so the condition $\Delta H=0$ is independent of the holomorphic chart.

= Regular covering map
{parent=Riemann surfaces}
{wiki=Covering_space}

A covering is regular when its deck group acts transitively on each fibre, equivalently when its fundamental-group subgroup is normal.

= Complete analytic function
{parent=Riemann surfaces}

A complete analytic function is the collection of all continuations of a germ. Its Riemann surface consists of these germs, with projection to their base points.

= Valency theorem
{parent=Riemann surfaces}
{wiki=Branched_covering}

For a nonconstant analytic map $f:R\to S$ between compact connected Riemann surfaces, there is an integer $\deg f$ such that for every $w\in S$,
$$
\sum_{z\in f^{-1}(w)}m_f(z)=\deg f.
$$

= Local degree of a holomorphic map
{title2=$m_f(p)$}
{parent=Valency theorem}

If a nonconstant holomorphic map has local-coordinate expression
$$
f(z)-f(p)=a(z-p)^m+O((z-p)^{m+1}),
\qquad a\ne0,
$$
then $m=m_f(p)$ is its local degree, multiplicity, or ramification index at $p$.

= Degree of a holomorphic map
{title2=$\deg f$}
{parent=Valency theorem}

For a nonconstant holomorphic map $f:X\to Y$ between compact connected Riemann surfaces, its degree is
$$
\deg f=\sum_{p\in f^{-1}(y)}m_f(p).
$$
The <valency theorem> says that this integer is independent of $y\in Y$.

= Degree of a rational map of the Riemann sphere
{parent=Valency theorem}

After cancelling common factors, the rational map $p/q$ on the Riemann sphere has degree $\max(\deg p,\deg q)$. It is an analytic isomorphism exactly when this degree is one, equivalently when it is a Möbius transformation.

= Degree of the derivative of a rational function
{title2=$\deg f'$}
{parent=Degree of a rational map of the Riemann sphere}

Let a nonconstant rational function $f$ have degree $d$, let $r$ be its number of distinct finite poles, and let $m_\infty$ be its pole order at infinity, taken as zero when infinity is not a pole. Differentiation raises each finite pole order by one and changes a pole of order $m_\infty>0$ at infinity into one of order $m_\infty-1$. Hence
$$
\deg f'=
\begin{cases}
d+r-1,&m_\infty>0,\\
d+r,&m_\infty=0.
\end{cases}
$$
In particular, $d-1\leq\deg f'\leq2d$.

= Degree of an elliptic function
{title2=$\deg g$}
{parent=Valency theorem}

The degree of a nonconstant elliptic function is the sum of the orders of its poles in a fundamental parallelogram. Equivalently, it is the degree of the induced holomorphic map from its complex torus to the Riemann sphere.

= Degree of the derivative of an elliptic function
{title2=$\deg g'$}
{parent=Degree of an elliptic function}

If an elliptic function $g$ has degree $d$ and $r$ distinct poles in a fundamental parallelogram, then each pole order increases by one under differentiation, so
$$
\deg g'=d+r.
$$
Because $1\leq r\leq d$, this gives $d+1\leq\deg g'\leq2d$.

= Octahedral rotation orbits on the Riemann sphere
{parent=Valency theorem}

The rotation group of the octahedron has order $24$. Its vertex, face-centre, and edge-centre stabilizers have orders $4,3,2$, giving exceptional orbit sizes $6,8,12$; every other orbit has size $24$.

= Degree-sized invariant separates finite-group orbits
{parent=Valency theorem}

Let a finite group $G$ of order $d$ act analytically on a compact Riemann surface, and let an invariant analytic map $F$ have degree $d$. At a point with stabilizer of order $e$, invariance forces the local multiplicity of $F$ to be at least $e$. Its orbit has $d/e$ points, so it already contributes at least $d$ to the fibre multiplicity. The valency theorem forces equality and shows that every fibre is exactly one orbit.

= Riemann-Hurwitz formula
{c}
{parent=Riemann surfaces}
{wiki=Riemann–Hurwitz_formula}

For a degree-$d$ holomorphic map, $2g_X-2=d(2g_Y-2)+\sum(e_p-1)$.

= Riemann sphere
{parent=Riemann surfaces}
{c}
{wiki}

The Riemann sphere is $\mathbb C\cup\{\infty\}$ with two charts related by reciprocal coordinates.

= Stereographic projection
{parent=Riemann sphere}
{wiki}

Stereographic projection identifies a sphere minus one pole with the plane and supplies the standard charts of the Riemann sphere.

= Local cyclic quotient of a Riemann surface
{parent=Riemann sphere}

Let a finite group $H$ of conformal automorphisms of the Riemann sphere fix $p$. The derivative representation at $p$ embeds $H$ into $\mathbb C^\times$, so $H$ is cyclic. Averaging a local coordinate linearizes the action to $z\mapsto\zeta z$, and the quotient has coordinate $u=z^{|H|}$.

= Finite conformal quotient of a Riemann surface
{parent=Local cyclic quotient of a Riemann surface}

For a finite conformal group action, choose a disc around each point that meets only its stabilizer translates. The local cyclic quotient chart $z\mapsto z^e$, where $e$ is the stabilizer order, gives the orbit space a Riemann-surface structure and makes the quotient map holomorphic.

= Orbit-separating invariant for the standard dihedral action on the Riemann sphere
{parent=Riemann sphere}

For $r(z)=\zeta z$ and $s(z)=1/z$, where $\zeta^n=1$, the rational function
$$
z^n+z^{-n}
$$
is invariant under $D_{2n}$. Equality of two values factors as $(u-v)(uv-1)=0$ for $u=z_1^n$ and $v=z_2^n$, so every fibre is exactly one dihedral orbit.

= Punctured Riemann surface
{parent=Riemann surfaces}

Removing a closed discrete set from a Riemann surface leaves an open complex one-manifold; if connected, it is again a Riemann surface.

= Path avoidance in a surface
{parent=Punctured Riemann surface}

A path in a surface can be perturbed inside coordinate discs to avoid finitely many prescribed points.

= Punctured complex plane
{parent=Punctured Riemann surface}
{wiki=Punctured_plane}

The punctured plane $\mathbb C^*$ is a connected Riemann surface homeomorphic to a cylinder.

= Cylinder as a punctured plane
{parent=Punctured complex plane}

Polar coordinates give $\mathbb C^*\cong S^1\times\mathbb R$ after taking logarithmic radius.

= Transport of a complex structure
{parent=Riemann surfaces}

A homeomorphism to a complex manifold transports its atlas and thereby defines a complex structure on the source.

= Nodal crossing
{parent=Riemann surfaces}
{wiki=Node_(algebraic_geometry)}

A nodal crossing locally consists of two complex branches meeting transversely and is not a one-dimensional complex manifold at the intersection.

= Topological manifold local obstruction
{parent=Nodal crossing}

If a punctured neighborhood has a different number of connected components from a punctured Euclidean ball, the point is not a manifold point.

= Reducible complex curve
{parent=Nodal crossing}

A reducible complex curve is a union of proper complex subcurves; intersecting components can create singular points.

= Elliptic integral of the first kind
{title2=$F(z,k)$}
{parent=Complex analysis}
{wiki=Elliptic_integral}

For a parameter $k$, one form of the incomplete elliptic integral of the first kind is
$$
F(z,k)=\int_0^z\frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}.
$$
Its value depends on the choices of square-root branches and, under <analytic continuation> around the <branch points>, on the integration path.

= Complete elliptic integral of the first kind
{title2=$K(k)$}
{parent=Elliptic integral of the first kind}
{wiki=Elliptic_integral#Complete_elliptic_integral_of_the_first_kind}

For $0<k<1$, the complete elliptic integral of the first kind is
$$
K(k)=\int_0^1\frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}}.
$$

= Complementary complete elliptic integral of the first kind
{title2=$K'(k)$}
{parent=Complete elliptic integral of the first kind}

For the complementary modulus $k'=\sqrt{1-k^2}$,
$$
K'(k)=K(k')
=\int_1^{1/k}\frac{dt}{\sqrt{(t^2-1)(1-k^2t^2)}}.
$$

= Elliptic function
{parent=Complex analysis}
{wiki=Elliptic_function}

An elliptic function is a <meromorphic function> with two real-linearly independent <periods> $\omega_1,\omega_2$. A <fundamental parallelogram of a period lattice>[fundamental cell] is a half-open parallelogram
$$
z_0+\{s\omega_1+t\omega_2:0\leq s,t<1\}.
$$
Opposite boundary integrals cancel. Consequently a nonconstant elliptic function has equally many zeros and poles in a cell, counted with multiplicity, and the sum of its pole residues in a cell is zero.

= Period lattice
{title2=$\Lambda$}
{parent=Elliptic function}
{wiki=Lattice_(group)}

Two real-linearly independent <periods> $\omega_1,\omega_2$ generate the period lattice
$$
\Lambda=\mathbb Z\omega_1+\mathbb Z\omega_2.
$$

= Fundamental parallelogram of a period lattice
{parent=Period lattice}

A fundamental parallelogram for $\Lambda=\mathbb Z\omega_1+\mathbb Z\omega_2$ is a translate of
$$
\{s\omega_1+t\omega_2:0\leq s,t<1\}.
$$
Its translates tile the <complex plane>.

= Fundamental parallelogram
{synonym}

= Nonconstant elliptic function has a pole
{parent=Elliptic function}

If an <elliptic function> had no <poles>, it would be <entire function>[entire]. It is bounded on the closure of a <fundamental parallelogram>, and periodicity then makes it bounded on the whole <complex plane>. The <Liouville theorem> would make it constant. Thus every nonconstant elliptic function has a pole.

= Jacobi elliptic sine
{title2=$\operatorname{sn}(u,k)$}
{parent=Elliptic function}
{c}
{wiki=Jacobi_elliptic_functions}

The Jacobi elliptic sine is the local inverse of the <elliptic integral of the first kind>: if
$$
u=\int_0^z\frac{dt}{\sqrt{(1-t^2)(1-k^2t^2)}},
$$
then $z=\operatorname{sn}(u,k)$. Its analytic continuation is a meromorphic doubly periodic function.

= Period lattice of the Jacobi elliptic sine
{parent=Jacobi elliptic sine}

For real $0<k<1$, the <period lattice> of $\operatorname{sn}(u,k)$ is generated by
$$
4K(k)
\quad\hbox{and}\quad
2iK'(k).
$$
The periods arise by composing the <monodromy reflection at a square-root branch point>[monodromy reflections] of its inverse <elliptic integral of the first kind>.

= Value multiplicity of an elliptic function
{parent=Elliptic function}

For any finite value $a$, apply the argument principle to $f-a$ around a fundamental parallelogram whose boundary avoids zeros and poles. Periodicity cancels the opposite-edge integrals, so the number of solutions of $f(z)=a$ equals the fixed number of poles of $f$, counting multiplicities.

= Weierstrass elliptic function
{title2=$\wp$}
{c}
{parent=Elliptic function}
{wiki=Weierstrass_elliptic_function}

The lattice sum $\wp(z)$ is an even elliptic function with a double pole of principal part $(z-\omega)^{-2}$ and zero residue at every lattice point $\omega$.

= Weierstrass zeta function
{title2=$\zeta$}
{c}
{parent=Weierstrass elliptic function}
{wiki=Weierstrass_zeta_function}

The Weierstrass zeta function satisfies $\zeta'=-\wp$ and is quasi-periodic: for every period $\omega$, the difference $\zeta(z+\omega)-\zeta(z)$ is constant. It has a simple pole of residue one at each lattice point. Consequently, if $\sum_jc_j=0$, then
$$
\sum_jc_j\zeta(z-a_j)
$$
is an elliptic function.

= Laurent coefficients of the Weierstrass elliptic function
{parent=Weierstrass elliptic function}

Writing $G_{2r}=\sum_{\omega\ne0}\omega^{-2r}$ for the lattice Eisenstein sums, expansion about zero gives
$$
\wp(z)=\frac1{z^2}
+\sum_{r=2}^{\infty}(2r-1)G_{2r}z^{2r-2}.
$$
Thus the constant coefficient vanishes and, for $k\geq1$, the coefficient of $z^{2k}$ is $(2k+1)G_{2k+2}$.

= Runge theorem
{c}
{parent=Complex analysis}
{wiki=Runge%27s_theorem}

A holomorphic function near a compact set can be uniformly approximated by rational functions with poles in prescribed complementary components; connected complement permits polynomials.

= Polynomial Runge theorem
{parent=Runge theorem}

If $K\subset\mathbb C$ is compact with connected complement and $f$ is holomorphic near $K$, then for each $\varepsilon>0$ there is a polynomial $p$ such that $\sup_K|p-f|<\varepsilon$.

= Polynomial approximation of the reciprocal on a proper circular arc
{parent=Polynomial Runge theorem}

A proper closed arc $S$ of a circle centered at zero has connected complement and stays away from zero. The polynomial Runge theorem therefore gives polynomials converging uniformly to $1/z$ on $S$.

= Pointwise approximation by a Runge exhaustion
{parent=Polynomial Runge theorem}

To approximate a piecewise constant function pointwise, exhaust each of its separated regions by compact subsets whose finite union has connected complement. Apply polynomial Runge approximation to the locally constant holomorphic function on each stage, with errors tending to zero. Every fixed point eventually lies in all later stages, so the uniform stagewise estimates imply pointwise convergence.

= Winding number
{title2=$\operatorname{wind}(\gamma,w)$}
{parent=Complex analysis}
{wiki}

For a closed piecewise smooth curve $\gamma$ avoiding $w$,
$$
\operatorname{wind}(\gamma,w)
=\frac1{2\pi i}\int_\gamma\frac{dz}{z-w}.
$$
It is the net change of a continuous argument divided by $2\pi$ and is an integer.

= Winding number of a continuous closed path
{parent=Winding number}

If $\gamma:[0,1]\to\mathbb C\setminus\{0\}$ is <continuous function>[continuous] and closed, choose a continuous lift $\theta$ such that
$$
\frac{\gamma(t)}{|\gamma(t)|}=e^{2\pi i\theta(t)}.
$$
Then $\operatorname{wind}(\gamma,0)=\theta(1)-\theta(0)\in\mathbb Z$. This agrees with the contour-integral definition when the path is piecewise smooth.

= Homotopy invariance of winding number
{parent=Complex analysis}

A homotopy through closed paths avoiding the base point cannot change the integer winding number.

= Dominated perturbation preserves winding number
{parent=Homotopy invariance of winding number}

If closed continuous paths $\gamma,\phi$ satisfy $|\gamma(t)|>|\phi(t)|$ for every $t$, then $H(s,t)=\gamma(t)+s\phi(t)$ never vanishes. It is therefore a homotopy through closed paths in $\mathbb C\setminus\{0\}$, and
$$
\operatorname{wind}(\gamma+\phi,0)=\operatorname{wind}(\gamma,0).
$$

= Winding-number proof of the fundamental theorem of algebra
{parent=Homotopy invariance of winding number}

For a polynomial $P(z)=a_nz^n+\cdots+a_0$ of positive degree, on a sufficiently large circle its leading term dominates the remaining terms. The <dominated perturbation preserves winding number>[dominated-perturbation lemma] therefore gives winding number $n$ to the loop $P(Re^{2\pi it})$. If $P$ had no zero, radial contraction of the input circle would map under $P$ to a homotopy with a constant loop in $\mathbb C\setminus\{0\}$, which has winding number zero. This contradiction proves the <Fundamental theorem of algebra>.

= Winding-number proof of the no-retraction theorem
{parent=Homotopy invariance of winding number}

If a continuous retraction from a closed disc to its boundary existed, applying it to a contraction of the boundary circle inside the disc would give a homotopy in $\mathbb C\setminus\{0\}$ from a loop of winding number one to a constant loop of winding number zero. <Homotopy invariance of winding number> rules this out.

= Bromwich contour
{parent=Complex analysis}

A Bromwich contour is a vertical line in the complex frequency plane lying to the right of the singularities in the inverse Laplace integral.

= Bromwich inversion with a square-root branch cut
{parent=Bromwich contour}

When an inverse Laplace integrand contains $p^{-1/2}$ on the principal branch, closing the Bromwich contour to the left encloses isolated poles and wraps a branch cut along the negative real axis. Pole residues give persistent oscillatory terms, while the jump across the two sides of the cut gives a real decaying integral.

= Branch point
{parent=Complex analysis}
{wiki}

A branch point is a point around which analytic continuation of a multivalued function returns a different value.

= Branch points
{synonym}

= Monodromy reflection at a square-root branch point
{parent=Branch point}

Let
$$
W(z)=\int_{z_0}^z\frac{q(t)}{\sqrt{P(t)}}\,dt,
$$
where $P$ has a simple zero at $a$, and let $A$ be the limiting value of $W$ at $a$ on one sheet. <Analytic continuation> once around $a$ changes the sign of the square root and hence of $W'$. The continued primitive agrees at $a$ with the original one, so it is
$$
W\longmapsto 2A-W.
$$

= Translation generated by two square-root monodromy reflections
{parent=Monodromy reflection at a square-root branch point}

If continuation around two square-root branch points acts on a primitive as $R_A(W)=2A-W$ and $R_B(W)=2B-W$, then
$$
R_B\circ R_A(W)=W+2(B-A).
$$
Thus pairs of branch-point loops generate additive periods of an inverse function.

= Complex number
{title2=$\mathbb C$}
{parent=Complex analysis}
{wiki}

A complex number has the form $x+iy$, with conjugate $x-iy$ and <modulus> $\sqrt{x^2+y^2}$.

= Real part
{title2=$\operatorname{Re}z$}
{parent=Complex number}
{wiki=Real_and_imaginary_parts}

For a <complex number> $z=x+iy$, its real part is $\operatorname{Re}z=x$.

= Imaginary part
{title2=$\operatorname{Im}z$}
{parent=Complex number}
{wiki=Real_and_imaginary_parts}

For a <complex number> $z=x+iy$, its imaginary part is $\operatorname{Im}z=y$.

= Euler's formula
{parent=Complex number}
{c}
{wiki=Euler%27s_formula}

Euler's formula states that $e^{i\theta}=\cos\theta+i\sin\theta$.

= Modulus
{title2=$|z|$}
{parent=Complex number}
{wiki=Absolute_value}

The modulus of a <complex number> $z=x+iy$ is $|z|=\sqrt{x^2+y^2}$.

= Moduli
{synonym}

= Complex conjugate
{title2=$\overline z$}
{parent=Complex number}
{wiki}

The complex conjugate of $z=x+iy$ is $\overline z=x-iy$. It satisfies $z\overline z=|z|^2$.

= Complex plane
{parent=Complex number}
{wiki}

The complex plane identifies the <complex number> $x+iy$ with the point $(x,y)$ in the real plane.

= Complex unit circle
{title2=$S^1$}
{parent=Complex plane}
{wiki=Unit_circle}

The complex unit circle is the subgroup $\{z\in\mathbb C:|z|=1\}$ under multiplication.

= Unit circle
{synonym}

= Isolated singularity
{parent=Complex analysis}
{wiki}

An isolated singularity at $a$ is a point at which a function is not holomorphic although it is holomorphic throughout some punctured neighbourhood of $a$.

= Classification of isolated singularities
{parent=Isolated singularity}
{wiki}

An isolated singularity is removable, a pole, or essential according as its Laurent principal part has zero, finitely many nonzero, or infinitely many nonzero terms.

= Removable singularity
{parent=Classification of isolated singularities}
{wiki}

An isolated singularity is removable when the function extends holomorphically across it, equivalently when its Laurent series has no negative-power terms.

= Riemann removable singularity theorem
{parent=Removable singularity}
{c}
{wiki=Riemann%27s_theorem_on_removable_singularities}

A holomorphic function bounded on a punctured neighbourhood extends holomorphically across the puncture.

= Removable singularity at infinity
{parent=Removable singularity}
{wiki}

A holomorphic function tending to a finite limit at infinity becomes holomorphic at zero after reciprocal substitution.

= Uniform L2 circle bound for a removable singularity
{parent=Removable singularity}

If $f$ is holomorphic on $0<|z|<R$ and
$$
\sup_{0<r<R}\int_0^{2\pi}|f(re^{i\theta})|^2\,d\theta<\infty,
$$
then the singularity at zero is removable. The Laurent coefficient formula and Cauchy--Schwarz give $|a_{-k}|=O(r^k)$ for each $k\geq1$, so every principal-part coefficient vanishes as $r\downarrow0$.

= Pole
{parent=Classification of isolated singularities}
{wiki=Zeros_and_poles}

A function has a pole of order $k$ at $a$ when $(z-a)^kf(z)$ extends holomorphically and is nonzero at $a$.

= Poles
{synonym}

= Meromorphic function
{parent=Pole}
{wiki}

A meromorphic function is <holomorphic function>[holomorphic] except at isolated <poles>. Equivalently, it is locally a quotient of two holomorphic functions whose denominator is not identically zero.

= Rational function
{parent=Meromorphic function}
{wiki}

A rational function is a quotient $p/q$ of <polynomials>, with $q$ nonzero. After cancelling common factors, it defines a <holomorphic map> from the <Riemann sphere> to itself.

= Partial fraction decomposition
{parent=Rational function}
{wiki}

Over a field in which the denominator splits, a proper rational function is a sum of terms $c/(x-a)^k$. The coefficient at a simple pole $a$ is obtained by multiplying by $x-a$ and evaluating at $a$.

= Order of vanishing
{title2=$\operatorname{ord}_p(f)$}
{parent=Rational function}
{wiki=Zeros_and_poles}

The order of vanishing of a nonzero <rational function> $f$ at a point $p$ is the exponent of a local parameter in its local factorization. A negative order is the order of a pole.

= Exponential of a pole is an essential singularity
{parent=Pole}

If $f$ has a pole at $a$, then $e^f$ has an essential singularity there. The exponential series produces infinitely many negative Laurent powers from the nonzero principal part of $f$.

= Essential singularity
{parent=Classification of isolated singularities}
{wiki}

An isolated singularity is essential when its Laurent series has infinitely many nonzero negative-power terms.

= Non-isolated singularity
{parent=Isolated singularity}
{wiki=Isolated_singularity}

A singularity is non-isolated when every punctured neighbourhood contains another singularity. An accumulation point of poles or essential singularities cannot itself be classified as a removable singularity, pole, or isolated essential singularity.

= Contour integration
{parent=Complex analysis}
{wiki}

Contour integration integrates complex functions along oriented curves and evaluates many real integrals through residues.

= Period obstruction to a holomorphic antiderivative
{parent=Contour integration}

A holomorphic function has an antiderivative on a domain only if its integral around every closed curve vanishes. Thus a single nonzero period, such as $\int_{|z|=1}dz/z=2\pi i$, prevents a global antiderivative.

= Jordan lemma
{parent=Contour integration}
{c}
{wiki=Jordan%27s_lemma}

Jordan's lemma controls exponential contour integrals on large semicircles and makes their arc contributions vanish.

= Complex line integral estimate
{parent=Contour integration}
{wiki}

Integrating along a straight segment bounds a complex integral by segment length times the supremum of the integrand.

= Hankel contour
{parent=Contour integration}
{c}
{wiki}

A Hankel contour runs along both banks of a branch cut and circles its branch point, converting the jump of a complex power into a sine factor.

= Hankel analytic continuation
{parent=Hankel contour}
{c}

Subtracting the local Taylor expansion at the encircled branch point extends a Hankel integral successively across left half-planes; prefactors often remove the introduced poles.

= Residue extraction by a Hankel contour
{parent=Hankel contour}

When the complex power becomes an integer power, a collapsed Hankel contour extracts the coefficient of $t^{-1}$ in the local Laurent series.

= Univalent function
{parent=Complex analysis}
{wiki}

A univalent function is a holomorphic injective function.

= Liouville theorem
{parent=Complex analysis}
{c}
{wiki}

Every bounded entire function is constant.

= Dense image of a nonconstant entire function
{parent=Liouville theorem}

The image of every nonconstant entire function is dense in $\mathbb C$. If a disc about $w$ were omitted, then $1/(f-w)$ would be bounded and entire, so Liouville's theorem would make $f$ constant.

= Entire function under a horizontal inverse-square-root bound
{parent=Liouville theorem}

If an entire function satisfies $|h(z)|\leq|\operatorname{Re}z|^{-1/2}$ away from the imaginary axis, then $h=0$. Cauchy's formula on $|z|=R$ bounds every Taylor coefficient by a constant times
$$
R^{-n-1/2}\int_0^{2\pi}|\cos\theta|^{-1/2}\,d\theta,
$$
which tends to zero as $R\to\infty$.

= Polynomial growth theorem for entire functions
{parent=Liouville theorem}
{wiki}

An entire function bounded by a polynomial in the radius is itself a polynomial, by Cauchy estimates.

= Cauchy derivative formula
{parent=Complex analysis}
{c}
{wiki}

Cauchy’s derivative formula expresses derivatives as contour integrals with higher-order Cauchy kernels.

= Lipschitz bound inside a bounded analytic half-plane
{parent=Cauchy derivative formula}

If $f$ is analytic and bounded by $K$ on $\operatorname{Re}z>0$, then Cauchy's derivative estimate gives
$$
|f'(z)|\leq K/c
$$
on $\operatorname{Re}z>c$. Integrating along line segments gives a Lipschitz bound with the same constant in that smaller half-plane.

= Locally uniform convergence of holomorphic functions
{parent=Complex analysis}
{wiki}

Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly.

= Schwarz reflection principle
{parent=Complex analysis}
{c}
{wiki}

A holomorphic function real on a boundary interval extends across it by conjugate reflection.

= Upper half-plane self-map
{parent=Complex analysis}
{wiki}

An upper-half-plane self-map is holomorphic and has positive imaginary part in the upper half-plane.

= Argument principle
{parent=Complex analysis}
{wiki}

If a meromorphic function has no zeros or poles on a positively oriented boundary $\gamma$, then
$$
\frac1{2\pi i}\int_\gamma\frac{f'}f\,dz=N-P,
$$
with zeros and poles counted by multiplicity.

= Rouché's theorem
{parent=Argument principle}
{c}
{wiki}

If $f,g$ are holomorphic near the closure of a bounded domain and $|g|<|f|$ on its boundary, then $f$ and $f+g$ have the same number of interior zeros, counted with multiplicity. Apply the argument principle to the zero-free boundary homotopy $f+tg$.

= Rouche theorem
{synonym}

= Hurwitz's theorem
{c}
{parent=Rouché's theorem}
{wiki=Hurwitz%27s_theorem_(complex_analysis)}

A locally uniform limit of nonvanishing holomorphic functions on a connected open set is either nonvanishing or identically zero. More generally, isolated zeros persist nearby with their multiplicity.

= Open mapping theorem
{disambiguate=complex analysis}
{parent=Rouché's theorem}
{wiki=Open_mapping_theorem_(complex_analysis)}

A nonconstant holomorphic function on a domain maps open sets to open sets. Around any point, isolate its zero relative to the value there and use <Rouché's theorem> to show that every sufficiently nearby value has a preimage.

= Maximum modulus principle from the complex open mapping theorem
{parent=open-mapping-theorem-complex-analysis}

If a nonconstant holomorphic function had a local maximum of its modulus, its open image near that point would contain values of larger modulus. Hence no such local maximum exists.

= Unit-disc image from a boundary modulus lower bound
{parent=Rouché's theorem}

If $f$ is holomorphic near the closed unit disc, $|f|\geq1$ on its boundary, and $|f(z_0)|<1$ at one interior point, then $f$ maps the open unit disc onto a set containing the open unit disc. Rouché's theorem first shows that $f$ has a zero and then that every $f-w$ with $|w|<1$ has one.

= Integer residue of a logarithmic derivative
{parent=Argument principle}

If $g$ is holomorphic and nonvanishing on a punctured disc, then
$$
\operatorname{res}_0\frac{g'}g
=\frac1{2\pi i}\int_C\frac{g'}g\,dz
$$
is the integer winding number of $g(C)$ about zero. If it equals $k$, the logarithmic derivative of $z^{-k}g(z)$ has a removable singularity at zero.

= Meromorphic function with prescribed zeros and poles
{parent=Complex analysis}
{wiki}

A meromorphic function with finitely many specified simple zeros and poles is a rational product up to a nonzero holomorphic factor.

= Maximum modulus principle
{parent=Complex analysis}
{wiki}

A nonconstant holomorphic function on a connected domain cannot attain a local maximum of its modulus at an interior point.

= Maximum modulus principle on a bounded domain
{parent=Maximum modulus principle}

If a function is continuous on the closure of a bounded plane domain and holomorphic inside, the compact closure supplies a point of maximum modulus. Unless the function is constant, the <maximum modulus principle> places every maximum on the boundary.

= Bounded half-plane maximum principle
{parent=Maximum modulus principle on a bounded domain}

Let $f$ be bounded and holomorphic on a half-plane and continuous on its closure. A bound $|f|\leq M$ on the boundary line propagates throughout the half-plane. Apply the bounded-domain principle to $f(z)z^{-1/n}$ on expanding truncated half-discs, then let $n\to\infty$.

= Mean value property for holomorphic functions
{parent=Complex analysis}
{wiki=Mean_value_property}

The value of a holomorphic function at the center of a closed disc equals its average around every concentric circle lying in the domain.

= Dirichlet beta function
{parent=Complex analysis}
{c}
{wiki}

The Dirichlet beta function is $\beta(s)=\sum_{n\geq0}(-1)^n(2n+1)^{-s}$ and is the Dirichlet L-function for the nontrivial character modulo four.

= Special values of the Dirichlet beta function
{parent=Dirichlet beta function}
{wiki=Dirichlet_beta_function\#Special_values}

Hankel residue extraction gives $\beta(0)=1/2$, $\beta(-2)=-1/2$, and values at other nonpositive integers from the Taylor coefficients of $1/(2\cosh t)$.

= Trivial zero of a Dirichlet L-function
{parent=Dirichlet beta function}
{wiki=Dirichlet_L-function\#Zeros}

Parity in the functional equation forces the beta function's trivial zeros $\beta(-1)=\beta(-3)=\cdots=0$.

= Dirichlet beta reflection formula
{parent=Dirichlet beta function}
{c}

The functional equation may be written
$$\beta(1-s)=\Gamma(s)(\pi/2)^{-s}\sin(\pi s/2)\beta(s).$$

= Identity theorem
{parent=Complex analysis}
{wiki=Identity_theorem}

Two holomorphic functions on a connected domain that agree on a set with an interior accumulation point agree everywhere.

= Residue at infinity from an asymptotic constant
{parent=Complex analysis}

If $F(z)=L+O(1/z)$ uniformly on large circles, then $F(z)/z$ has residue $L$ at infinity in the large-contour sense and its counterclockwise large-circle integral tends to $2\pi iL$.

= Large-circle contour estimate
{parent=Complex analysis}

On a circle of radius $R$, an integrand uniformly of order $R^{-2}$ has contour integral of order $R^{-1}$ by the estimation lemma.

= Fuchsian differential equation
{parent=Complex analysis}
{c}
{wiki}

A Fuchsian differential equation has only regular singular points, including possibly the point at infinity.

= Ordinary point criterion for a second-order equation
{parent=Fuchsian differential equation}

For $y''+P(z)y'+Q(z)y=0$, the finite point $z_0$ is ordinary when $P$ and $Q$ are <holomorphic function>[holomorphic] at $z_0$.

= Characteristic exponent at a regular singular point
{parent=Fuchsian differential equation}

Substitution of a Frobenius behavior $(z-z_0)^\rho$ into the leading singular terms gives the indicial equation and its characteristic exponents.

= Regular singular point criterion for a second-order equation
{parent=Fuchsian differential equation}

For
$$
y''+P(z)y'+Q(z)y=0,
$$
$z_0$ is a regular singular point when $(z-z_0)P(z)$ and $(z-z_0)^2Q(z)$ are analytic at $z_0$.

= Regular singular point at infinity
{parent=Regular singular point criterion for a second-order equation}

Under $\zeta=1/z$, the point $z=\infty$ is regular singular for
$$
y''+P(z)y'+Q(z)y=0
$$
exactly when $zP(z)$ and $z^2Q(z)$ are <holomorphic function>[holomorphic] functions of $1/z$ near $1/z=0$.

= Logarithmic solution from a repeated Frobenius exponent
{parent=Regular singular point criterion for a second-order equation}

When a second-order equation has a repeated indicial root and only one independent Frobenius series, a second solution has the local form
$$
y_2(z)=y_1(z)\log(z-z_0)+\text{a Frobenius series}.
$$

= Irregular singular point
{parent=Fuchsian differential equation}
{wiki=Regular_singular_point}

A singular point of a linear differential equation is irregular when it fails the regular-singular criterion. For $y''+P(x)y'+Q(x)y=0$, a pole of order greater than one in $P$ or greater than two in $Q$ makes the point irregular.

= Papperitz symbol
{parent=Fuchsian differential equation}
{c}
{wiki=Riemann%27s_differential_equation}

A Papperitz or P-symbol lists the three regular singular points of a second-order equation and the two characteristic exponents at each.

= Fuchs relation
{parent=Papperitz symbol}
{c}
{wiki=Fuchs%27_relation}

For a second-order equation with three regular singular points, the sum of all six characteristic exponents is one.

= Möbius transformation of a Papperitz symbol
{parent=Papperitz symbol}
{c}

A Möbius change of independent variable permutes the singular points while carrying their characteristic exponent pairs with them.

= Dependent-variable rescaling of a Papperitz symbol
{parent=Papperitz symbol}

Multiplying a solution by $(z-z_0)^\lambda$ adds $\lambda$ to both exponents at the finite point $z_0$ and subtracts $\lambda$ from both exponents at infinity, under the convention that an exponent $\rho$ at infinity corresponds to behavior $z^{-\rho}$.

= Gauss hypergeometric equation
{parent=Fuchsian differential equation}
{c}
{wiki=Hypergeometric_function}

The Gauss equation has singularities at $0,1,\infty$ and normalized exponent-zero solution $F(A,B;C;u)$ at the origin.

= Pfaff transformation
{c}
{parent=Gauss hypergeometric equation}
{wiki=Hypergeometric_function#Transformation_formulas}

With compatible branches and initially near $z=0$,
$$
F\left(a,c-b;c;\frac{z}{z-1}\right)
=(1-z)^aF(a,b;c;z).
$$
It follows by applying a Möbius transformation and a dependent-variable rescaling to the same <Papperitz symbol>, then matching the normalized analytic solution at zero.

= Second local hypergeometric solution
{parent=Gauss hypergeometric equation}

When $C$ is not an integer, a second local solution at zero is $u^{1-C}F(A-C+1,B-C+1;2-C;u)$.

= Hypergeometric connection formula at infinity
{parent=Gauss hypergeometric equation}

Away from resonant parameter cases, two independent local solutions at infinity are
$$
z^{-a}F(a,1+a-c;1+a-b;z^{-1})
$$
and the expression obtained by exchanging $a$ and $b$. Every analytic continuation of a hypergeometric solution into their common domain is a constant linear combination of this basis.

= Hypergeometric cancellation identity
{parent=Gauss hypergeometric equation}

The binomial series gives $F(A,C;C;u)=(1-u)^{-A}$ wherever the defining series converges, followed by analytic continuation.

= Elementary specialization of a hypergeometric solution
{parent=Hypergeometric cancellation identity}

Special parameter choices can identify a hypergeometric solution with an elementary solution of the same second-order differential equation by matching its local normalization.

= Laplace transform
{title2=$\mathcal L$}
{parent=Analysis}
{wiki}

The Laplace transform is $\widehat f(p)=\int_0^\infty e^{-pt}f(t)dt$ and converts differentiation into multiplication by $p$ plus initial data.

= Laplace transform of a derivative
{parent=Laplace transform}
{c}

Integration by parts gives
$$
\mathcal L\{f'\}(s)=s\widehat f(s)-f(0)
$$
whenever the boundary term at infinity vanishes.

= Laplace transform shift theorem
{parent=Laplace transform}
{c}
{wiki=Laplace_transform\#Time_shifting}

For $a\geq0$, $\mathcal L\{f(t-a)H(t-a)\}(s)=e^{-as}\mathcal L\{f\}(s)$.

= Laplace transform time-shift rule
{parent=Laplace transform shift theorem}
{c}

The delayed unit step has
$$
\mathcal L\{H(t-t_0)\}(s)=\frac{e^{-st_0}}s,
\qquad \operatorname{Re}s>0.
$$

= Heaviside step function
{title2=$H$}
{parent=Laplace transform shift theorem}
{c}
{wiki}

The Heaviside function is zero before its switching time and one after it, allowing delayed signals to be written algebraically.

= Laplace transform of a periodic function
{parent=Laplace transform}
{c}

If $g$ has period $T$, then
$$\mathcal L\{g\}(s)=\frac{\int_0^T e^{-st}g(t)dt}{1-e^{-sT}}.$$

= Matrix-valued Laplace transform
{parent=Laplace transform}

The Laplace transform of a matrix-valued function is obtained by transforming each entry. A constant matrix may be taken outside on the corresponding side of the integral; in particular, $\mathcal L\{AB\}=A\mathcal L\{B\}$ when $A$ is constant.

= Laplace-transform solution of a constant-coefficient vector ODE
{parent=Matrix-valued Laplace transform}

For
$$
y'=Ay+g,qquad y(0)=y_0,
$$
the derivative rule gives
$$
(sI-A)Y(s)=y_0+G(s),
$$
and hence $Y(s)=(sI-A)^{-1}(y_0+G(s))$ wherever the transforms converge and $sI-A$ is invertible.

= Hardy-Littlewood maximal function
{title2=$Mf$}
{c}
{parent=Analysis}
{wiki=Hardy%E2%80%93Littlewood_maximal_function}

The centered Hardy-Littlewood maximal function on the real line is
$$
Mf(x)=\sup_{r>0}\frac1{2r}\int_{x-r}^{x+r}|f(t)|\,dt.
$$

= Hardy-Littlewood maximal inequality
{c}
{parent=Hardy-Littlewood maximal function}
{wiki=Hardy–Littlewood_maximal_function}

The maximal averaging operator has weak type $(1,1)$, controlling the measure of points where local averages are large.

= Watson lemma
{c}
{parent=Analysis}
{wiki=Watson%27s_lemma}

Watson lemma obtains an endpoint Laplace-integral asymptotic expansion by integrating the local power expansion of its amplitude term by term.

= Reciprocal-amplitude Laplace integral
{parent=Watson lemma}

Expanding $(1+t)^{-1}$ at the endpoint and applying Watson's lemma gives
$$
\int_0^\infty\frac{e^{-xt}}{1+t}\,dt
\sim\sum_{n=0}^\infty(-1)^n n!x^{-n-1}
\qquad(x\to+\infty).
$$

= Incomplete gamma function
{parent=Analysis}
{wiki=Incomplete_gamma_function}

The upper incomplete gamma function is $\gamma(x,y)=\int_y^\infty t^{x-1}e^{-t}dt$ under the convention used here.

= Asymptotic expansion by repeated integration by parts
{parent=Incomplete gamma function}

Repeated integration by parts generates successive inverse powers of a large endpoint, with a remainder expressed by the same integral with shifted parameters.

= Large-lower-limit expansion of the incomplete gamma function
{parent=Asymptotic expansion by repeated integration by parts}

For fixed $x$ and $y\to\infty$,
$$
\gamma(x,y)\sim y^{x-1}e^{-y}
\sum_{n\geq0}\left(\prod_{j=1}^n(x-j)\right)y^{-n}.
$$

= Fixed-lower-limit sine-integral expansion
{parent=Asymptotic expansion by repeated integration by parts}

For
$$
\operatorname{si}(x)=\int_1^\infty\frac{\sin(xt)}t\,dt,
$$
repeated integration by parts gives
$$
\operatorname{si}(x)\sim
\cos x\sum_{n\geq0}(-1)^n(2n)!x^{-2n-1}
+\sin x\sum_{n\geq0}(-1)^n(2n+1)!x^{-2n-2}.
$$

= Fixed-lower-limit incomplete gamma asymptotic
{parent=Incomplete gamma function}

For fixed finite $y$ and $x\to\infty$, the omitted integral from zero to $y$ is negligible, so $\gamma(x,y)\sim\Gamma(x)$ and Stirling's formula applies.

= Endpoint expansion of the incomplete gamma function
{parent=Incomplete gamma function}

When $y=\lambda x$ with $\lambda>1$, the integral is dominated by its lower endpoint. Setting $t=y+s$ and applying Watson's lemma gives coefficients rational in $\lambda-1$.

= Stationary phase method
{parent=Analysis}
{wiki}

Oscillatory integrals are led by points where the phase derivative vanishes, with a phase shift determined by the Hessian signature.

= Nondegenerate stationary point of a phase
{parent=Stationary phase method}

A <stationary point> $x_0$ of a <smooth function>[smooth] phase $\phi$ is nondegenerate when its <Hessian matrix> at $x_0$ is an <invertible matrix>. In one dimension, this condition is $\phi''(x_0)\ne0$.

= Nondegenerate stationary points of a phase
{synonym}

= One-dimensional stationary-phase formula
{parent=Stationary phase method}

If $\psi'(k_0)=0$, $\psi''(k_0)\ne0$, and $A$ is smooth near $k_0$, then
$$
\int A(k)e^{it\psi(k)}\,dk
\sim A(k_0)e^{it\psi(k_0)+i\pi\operatorname{sgn}(\psi''(k_0))/4}
\sqrt{\frac{2\pi}{t|\psi''(k_0)|}}.
$$
Contributions from distinct stationary points add.

= Nonstationary oscillatory integral
{parent=Stationary phase method}

If the phase derivative has no zero on the support of a smooth amplitude, repeated <integration by parts> makes the oscillatory integral decay faster than any prescribed inverse power, subject to corresponding decay and regularity of the amplitude.

= WKB approximation
{c}
{parent=Analysis}
{wiki=WKB_approximation}

For a large-parameter second-order ODE, WKB separates a rapidly varying exponential phase from a slowly varying transport amplitude.

= Airy turning-point connection formula
{parent=WKB approximation}
{c}

Near a simple turning point, rescaling the independent variable reduces a second-order equation to the Airy equation. The decaying Airy solution on the forbidden side matches an oscillatory solution with phase shift $\pi/4$ on the allowed side.

= Classical turning point
{parent=WKB approximation}

For a one-dimensional energy $E$ and potential $V(x)$, a classical turning point satisfies $V(x)=E$. The classically allowed region has $E>V(x)$ and supports an oscillatory leading-order WKB solution.

= WKB quantization condition
{parent=WKB approximation}
{c}

For an oscillatory interval terminated by a hard boundary and a simple turning point, matching the boundary condition to the decaying solution gives an action integral equal to an integer multiple of $\pi$ with the turning-point phase correction.

= Two-turning-point WKB quantization condition
{parent=WKB quantization condition}

For two simple <classical turning point>[turning points] $a(E)<b(E)$ surrounding one classically allowed well,
$$
\frac1\varepsilon\int_{a(E)}^{b(E)}\sqrt{E-V(x)}\,dx
=\left(n+\frac12\right)\pi,
\qquad n=0,1,2,\ldots.
$$

= Semiclassical action integral
{title2=$f(E)$}
{parent=Two-turning-point WKB quantization condition}

The semiclassical action integral
$$
f(E)=\int_{a(E)}^{b(E)}\sqrt{E-V(x)}\,dx
$$
is strictly increasing for a confining one-dimensional potential. Consequently the <two-turning-point WKB quantization condition> has at most one energy at each quantum number.

= Action integral for a linear-to-square-root potential
{parent=Semiclassical action integral}

For the even potential
$$
V(x)=\begin{cases}|x|/4,&|x|\leq4,\\
\sqrt{|x|}-1,&|x|\geq4,
\end{cases}
$$
the turning points are $\pm4E$ for $0\leq E\leq1$ and $\pm(E+1)^2$ for $E\geq1$. Its action is
$$
f(E)=\begin{cases}
\dfrac{16}{3}E^{3/2},&0\leq E\leq1,\\[4pt]
\dfrac{16}{3}E^{3/2}+\dfrac{16}{15}(E-1)^{5/2},&E\geq1.
\end{cases}
$$

= Removal of the first derivative from a second-order differential equation
{parent=WKB approximation}

For
$$
y''+P(x)y'+Q(x)y=0,
$$
the substitution
$$
y=e^{-\frac12\int P(x)\,dx}v
$$
gives the normal form
$$
v''+\left(Q-\frac12P'-\frac14P^2\right)v=0.
$$

= Irregular singular point at infinity
{parent=WKB approximation}
{wiki=Regular_singular_point}

To classify infinity, set $z=1/x$ and classify $z=0$. For a second-order equation in normal form, a transformed coefficient with a pole of order greater than two makes infinity an irregular singular point.

= Liouville-Green exponential ansatz
{parent=WKB approximation}
{c}

Writing $v=e^S$ in $v''=q(x)v$ produces the <Riccati equation>
$$
S''+(S')^2=q(x).
$$
Expanding $S'$ in inverse powers of $x$ determines the exponential phase and algebraic prefactor recursively.

= Riccati equation
{parent=Liouville-Green exponential ansatz}
{c}
{wiki}

A Riccati equation is a first-order equation quadratic in the unknown function.

= Numerical analysis
{parent=Analysis}
{wiki}

Numerical analysis studies algorithms for approximating mathematical problems and controlling their errors.

= Arithmetic operation
{parent=Numerical analysis}

An arithmetic operation is one scalar addition, subtraction, multiplication, or division. Counting such operations gives a basic model of an algorithm's computational cost.

= Schur stability criterion
{parent=Numerical analysis}
{c}
{wiki=Jury_stability_criterion}

For a real monic quadratic $z^2+a_1z+a_2$, both roots lie in the closed unit disk when
$$
|a_2|\leq1,
\qquad
1+a_1+a_2\geq0,
\qquad
1-a_1+a_2\geq0,
$$
with the usual simplicity requirement for roots on the unit circle.

= Lie-Trotter splitting commutator error
{parent=Numerical analysis}
{c}
{wiki=Lie_product_formula}

For bounded matrices,
$$
e^{kB}e^{kC}=e^{k(B+C)}+\frac{k^2}{2}(BC-CB)+O(k^3).
$$
Commutativity removes this splitting error, though first-order substeps such as backward Euler retain their own quadratic local errors.

= Composite midpoint rule error
{parent=Numerical analysis}

For $f\in C^2[a,b]$, the composite midpoint rule with mesh width $h$ has local error $O(h^3)$ and global error $O(h^2)$.

= Quadratic substitution for a square-root endpoint singularity
{parent=Composite midpoint rule error}

If $f(x)=x^{-1/2}g(x)$ with $g$ smooth near zero, the substitution $x=s^2$ gives
$$
\int_0^1f(x)\,dx=\int_0^12g(s^2)\,ds.
$$
The transformed integrand is smooth, so an ordinary second-order midpoint rule recovers its usual convergence rate.

= Polynomial interpolation
{parent=Numerical analysis}
{wiki}

Given distinct nodes $x_0,\ldots,x_n$ and values $f(x_i)$, there is a unique polynomial of degree at most $n$ taking those values.

= Lagrange interpolation polynomial
{c}
{parent=Polynomial interpolation}
{wiki=Lagrange_polynomial}

The interpolating polynomial is
$$
p_n(x)=\sum_{j=0}^nf(x_j)
\prod_{\substack{0\leq i\leq n\\i\ne j}}
\frac{x-x_i}{x_j-x_i}.
$$

= Divided difference
{parent=Polynomial interpolation}
{wiki}

For distinct nodes,
$$
f[x_0,\ldots,x_k]
=\sum_{j=0}^k\frac{f(x_j)}
{\prod_{i\ne j}(x_j-x_i)}.
$$
It is the leading coefficient of the interpolating polynomial through those nodes and obeys
$$
f[x_0,\ldots,x_k]
=\frac{f[x_1,\ldots,x_k]-f[x_0,\ldots,x_{k-1}]}
{x_k-x_0}.
$$

= Newton interpolation polynomial
{c}
{parent=Divided difference}
{wiki=Newton_polynomial}

Newton's nested interpolation form is
$$
p_n(x)=f(x_0)+\sum_{k=1}^nf[x_0,\ldots,x_k]
\prod_{i=0}^{k-1}(x-x_i).
$$
A triangular divided-difference table computes all its coefficients using exactly $n(n+1)/2$ divisions.

= Discrete Fourier transform
{parent=Numerical analysis}
{wiki}

For samples $y_0,\ldots,y_{m-1}$, their discrete Fourier transform is
$$Y_k=\sum_{n=0}^{m-1}y_ne^{-2\pi ink/m}.$$

= Cooley--Tukey fast Fourier transform
{parent=Discrete Fourier transform}
{c}
{wiki=Cooley–Tukey_FFT_algorithm}

The radix-two Cooley--Tukey algorithm recursively splits a discrete Fourier transform into transforms of the even-indexed and odd-indexed samples. For a power-of-two length this evaluates the transform in $O(m\log m)$ arithmetic operations.

= Even--odd fast Fourier transform recursion
{parent=Cooley--Tukey fast Fourier transform}

For $\omega_m=e^{-2\pi i/m}$ and length-$m/2$ transforms $E_k,O_k$ of the even and odd samples,
$$Y_k=E_k+\omega_m^kO_k,
\qquad Y_{k+m/2}=E_k-\omega_m^kO_k.$$

= Discrete cosine transform
{parent=Discrete Fourier transform}
{wiki}

The type-II discrete cosine transform of $x_0,\ldots,x_{N-1}$ is
$$Z_k=\sum_{n=0}^{N-1}x_n\cos\left(\frac{\pi}{N}\left(n+\frac12\right)k\right).$$

= Discrete cosine transform from an even-reflected discrete Fourier transform
{parent=Discrete cosine transform}

Reflect $x$ evenly to length $2N$ by $y_n=x_n$ and $y_{2N-1-n}=x_n$. If $Y$ is the discrete Fourier transform of $y$, then
$$Z_k=\frac12e^{-\pi ik/(2N)}Y_k.$$
This reduction computes the cosine transform with one fast Fourier transform and $O(N)$ additional work.

= Discrete sine transform
{parent=Discrete Fourier transform}
{wiki}

One half-grid sine-transform convention is
$$\widetilde Z_k=\sum_{n=0}^{N-1}x_n
\sin\left(\frac{\pi}{N}\left(n+\frac12\right)(k+1)\right).$$

= Discrete sine transform from a sign-modulated discrete cosine transform
{parent=Discrete sine transform}

If $\xi_n=(-1)^nx_n$ and $C_j(\xi)$ is its type-II cosine transform, then
$$\widetilde Z_k=C_{N-1-k}(\xi).$$
Thus a sign modulation, one fast cosine transform, and output reversal give an $O(N\log N)$ sine-transform algorithm.

= Gershgorin circle theorem
{c}
{parent=Numerical analysis}
{wiki=Gershgorin_circle_theorem}

Every eigenvalue of a complex matrix $A=(a_{ij})$ lies in at least one disk
$$
\left\{z:|z-a_{ii}|\leq\sum_{j\ne i}|a_{ij}|\right\}.
$$

= Gershgorin stability bound for a variable-coefficient diffusion stencil
{parent=Gershgorin circle theorem}

A symmetric conservative five-point diffusion matrix has row center $-s/h^2$ and radius at most $s/h^2$, where $s$ is the sum of its four nonnegative edge coefficients. If every coefficient is at most $a_{\max}$, its eigenvalues lie in $[-8a_{\max}/h^2,0]$. Forward Euler is therefore stable when $\Delta t/h^2\leq1/(4a_{\max})$.

= Finite difference method
{parent=Numerical analysis}
{wiki}

A finite difference method replaces derivatives by algebraic combinations of values on a discrete space-time grid.

= Central finite difference
{parent=Finite difference method}
{wiki=Finite_difference_coefficient}

The centered unit-step approximation to a second derivative is
$$
f''(0)\approx f(-1)-2f(0)+f(1).
$$

= Forward difference operator
{title2=$\Delta$}
{parent=Finite difference method}
{wiki=Finite_difference}

The forward difference of a sequence or function on the <integer>[integers] is
$$
(\Delta f)(n)=f(n+1)-f(n).
$$

= Discrete antiderivative
{parent=Forward difference operator}

A discrete antiderivative of $f$ is a function $F$ satisfying $\Delta F=f$. <Pascal's identity> gives
$$
\Delta\binom nr=\binom n{r-1}.
$$

= Newton series for an integer-valued polynomial sequence
{parent=Forward difference operator}
{c}

If $\Delta^{k+1}f=0$, then $f$ is an integer linear combination of $1,\binom n1,\ldots,\binom nk$ whenever $f$ is integer-valued. Successive differences recover the coefficients.

= Von Neumann stability analysis
{parent=Finite difference method}
{c}
{wiki=Von_Neumann_stability_analysis}

Von Neumann analysis inserts the Fourier mode $u_m^n=G^n e^{im\theta}$ into a constant-coefficient difference scheme. A one-step method is stable in the discrete $2$-norm when its amplification factor satisfies $|G(\theta)|\leq1$ for every resolvable wavenumber.

= Amplification factor
{title2=$H(\theta)$}
{parent=Von Neumann stability analysis}

For a one-step translation-invariant recurrence, the amplification factor is the multiplier satisfying
$$
\widehat u^{\,n+1}(\theta)=H(\theta)\widehat u^{\,n}(\theta).
$$
The <Parseval identity> converts the pointwise bound $|H|\leq1$ into non-growth of the discrete $2$-norm.

= Amplification factor of a two-sided one-step stencil
{parent=Amplification factor}

For
$$
\sum_{k=r}^sa_ku^{n+1}_{m+k}
=\sum_{k=r}^sb_ku^n_{m+k},
$$
the Fourier convention $\widehat u(\theta)=\sum_me^{-im\theta}u_m$ gives
$$
H(\theta)=
\frac{\sum_{k=r}^sb_ke^{ik\theta}}
{\sum_{k=r}^sa_ke^{ik\theta}},
$$
provided the denominator does not vanish.

= Amplification polynomial of a multilevel finite difference scheme
{parent=Von Neumann stability analysis}

For a scheme
$$
\sum_{r,j}a_{rj}u_{m+j}^{n+r}=0,
$$
the Fourier ansatz gives the amplification polynomial
$$
\sum_{r,j}a_{rj}G^r e^{ij\theta}=0.
$$
Its roots are the amplification factors of that Fourier mode.

= Crank-Nicolson diffusion scheme
{parent=Von Neumann stability analysis}
{c}
{wiki=Crank%E2%80%93Nicolson_method}

For the centered spatial second difference, the Crank-Nicolson diffusion scheme has amplification factor
$$
G(\theta)=\frac{1-2\mu\sin^2(\theta/2)}
{1+2\mu\sin^2(\theta/2)}.
$$
It is stable for every $\mu\geq0$.

= Centered three-level wave scheme
{parent=Von Neumann stability analysis}

For
$$
v_m^{n+1}-2\rho v_m^n+v_m^{n-1}
=\mu(v_{m+1}^n-2v_m^n+v_{m-1}^n),
$$
the amplification polynomial is
$$
G^2+(4\mu\sin^2(\theta/2)-2\rho)G+1=0.
$$
All modes have unit-modulus roots only when $\rho=1$ and $0\leq\mu\leq1$.

= Courant number
{parent=Finite difference method}
{c}
{wiki=Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition}

The Courant number is the dimensionless ratio of physical propagation during one time step to one spatial grid spacing, commonly $\mu=c\Delta t/\Delta x$.

= Dirichlet discrete Laplacian
{parent=Finite difference method}

On $J$ interior points, the one-dimensional centered second-difference matrix with homogeneous Dirichlet boundaries has eigenvalues
$$
\lambda_j=-4\sin^2\left(\frac{j\pi}{2(J+1)}\right),
\qquad 1\leq j\leq J.
$$

= Crank-Nicolson stability on a finite Dirichlet interval
{parent=Dirichlet discrete Laplacian}

If $L$ is the negative semidefinite Dirichlet discrete Laplacian, the Crank-Nicolson amplification matrix
$$
Q=(I-\mu L/2)^{-1}(I+\mu L/2)
$$
has eigenvalues $(1+\mu\lambda_j/2)/(1-\mu\lambda_j/2)$ of modulus at most one for every $\mu\geq0$.

= Five-point Dirichlet Laplacian as a Kronecker sum
{parent=Dirichlet discrete Laplacian}

If $T$ is the one-dimensional Dirichlet second-difference matrix on $m$ interior points, the two directional parts of the two-dimensional five-point Laplacian are
$$
A_x=T\otimes I_m,
\qquad
A_y=I_m\otimes T.
$$
They commute, and $A_x+A_y$ is the Kronecker-sum discretization of the Laplacian.

= One-implicit-direction diffusion splitting
{parent=Five-point Dirichlet Laplacian as a Kronecker sum}

The split step
$$
(I-\mu A_y)u^{n+1/2}=u^n,
\qquad
u^{n+1}=(I+\mu A_x)u^{n+1/2}
$$
has amplification matrix
$$
C=(I+\mu A_x)(I-\mu A_y)^{-1}.
$$
On a common directional eigenvector its eigenvalue is $(1+\mu\lambda_p)/(1-\mu\lambda_q)$.

= Stability limit of one-implicit-direction diffusion splitting
{parent=One-implicit-direction diffusion splitting}

On the $m$ by $m$ Dirichlet grid with $h=1/(m+1)$, the exact discrete stability condition for $m>1$ is
$$
0<\mu\leq\frac1{2\cos(\pi h)}.
$$
The mesh-independent sufficient condition is $0<\mu\leq1/2$, and the exact bound tends to $1/2$ as $h\to0$.

= Unconditionally stable corrected directional diffusion splitting
{parent=One-implicit-direction diffusion splitting}

Adding the correction
$$
u^{n+1}=\widetilde u^{n+1}+\mu A_x(u^{n+1}-u^n)
$$
gives
$$
D=(I-\mu A_x)^{-1}(I+\mu^2A_xA_y)(I-\mu A_y)^{-1}.
$$
Its common-basis eigenvalues are
$$
d_{pq}=\frac{1+\mu^2\lambda_p\lambda_q}
{(1-\mu\lambda_p)(1-\mu\lambda_q)}.
$$
Since $\lambda_p,\lambda_q<0$, one has $0<d_{pq}\leq1$ for every $\mu>0$.

= Thomas algorithm
{parent=Numerical analysis}
{c}
{wiki=Tridiagonal_matrix_algorithm}

The Thomas algorithm is specialized Gaussian elimination for a tridiagonal linear system and uses linear time and storage in the system dimension.

= Gaussian quadrature
{parent=Numerical analysis}
{wiki}

An $n$-node Gaussian quadrature rule uses orthogonal-polynomial roots and integrates every polynomial of degree at most $2n-1$ exactly.

= Chebyshev polynomial
{c}
{parent=Numerical analysis}
{wiki}

$T_n(\cos\theta)=\cos(n\theta)$. These polynomials are orthogonal for weight $(1-x^2)^{-1/2}$ and support spectrally accurate approximation.

= Chebyshev polynomial of the second kind
{c}
{parent=Chebyshev polynomial}
{wiki}

The Chebyshev polynomial of the second kind is defined by
$$
U_n(\cos\theta)=\frac{\sin((n+1)\theta)}{\sin\theta}.
$$
It is orthogonal on $[-1,1]$ with weight $\sqrt{1-x^2}$ and has roots $\cos(j\pi/(n+1))$ for $1\leq j\leq n$.

= Rational cosine of an integral submultiple of pi
{parent=Chebyshev polynomial}

The number $2\cos(\pi/n)$ is an algebraic integer. If it is rational it must be an integer, which shows that $\cos(\pi/n)$ is rational only for $n=1,2,3$.

= Positivity of Chebyshev derivatives beyond the unit interval
{parent=Chebyshev polynomial}

Every root of $T_n^{(k)}$ lies in $(-1,1)$ and its leading coefficient is positive. Hence
$$
T_n^{(k)}(x)>0
$$
for $x\geq1$ and $0\leq k\leq n$.

= Monic Chebyshev extremal polynomial
{parent=Chebyshev polynomial}

Among monic real polynomials of degree $n$, $2^{1-n}T_n$ uniquely minimizes the supremum norm on $[-1,1]$, attaining the value $2^{1-n}$.

= Chebyshev polynomial domination lemma
{parent=Chebyshev polynomial}
{c}

If $f$ has degree at most $n$ and $|f|<1$ on $[-1,1]$, alternation at the extrema of $T_n$ implies $|f(t)|<|T_n(t)|$ outside that interval.

= Chebyshev differential equation
{parent=Chebyshev polynomial}
{c}
{wiki}

The equation $(1-z^2)w''-zw'+n^2w=0$ is obtained from the harmonic equation $w_{xx}+n^2w=0$ by $z=\cos x$.

= Chebyshev derivative Sturm-Liouville pair
{c}
{parent=Chebyshev differential equation}

The Chebyshev equation has self-adjoint form
$$
\left(\sqrt{1-x^2}\,T_n'\right)'
+\frac{n^2}{\sqrt{1-x^2}}T_n=0.
$$
For $U_n=T_n'$, differentiation gives
$$
\left((1-x^2)^{3/2}U_n'\right)'
+(n^2-1)\sqrt{1-x^2}\,U_n=0.
$$
The respective orthogonality weights are $(1-x^2)^{-1/2}$ and $(1-x^2)^{1/2}$.

= Backward differentiation formula
{parent=Numerical analysis}
{wiki}

= Constrained optimization
{parent=Numerical analysis}
{wiki}

= Equally spaced interpolation
{parent=Numerical analysis}
{wiki}

= Ford-Fulkerson algorithm
{parent=Numerical analysis}
{c}
{wiki}

= Givens rotation
{parent=Numerical analysis}
{c}
{wiki}

A Givens rotation is the identity except for a coordinate-plane block
$$
\begin{pmatrix}c&s\\-s&c\end{pmatrix},
\qquad c^2+s^2=1.
$$
Choosing $(c,s)=(a,b)/\sqrt{a^2+b^2}$ maps the vector $(a,b)^T$ to $(\sqrt{a^2+b^2},0)^T$.

= QR decomposition by Givens rotations
{c}
{parent=Givens rotation}

Successive Givens rotations eliminate entries below a matrix diagonal. If their product is $\Omega$ and $R=\Omega A$ is upper triangular, then
$$
A=QR,
\qquad Q=\Omega^T,
$$
is a QR decomposition.

= Gradient descent
{parent=Numerical analysis}
{wiki}

Gradient descent updates $x$ in the negative-gradient direction. For $F(x)=x^TAx/2-b^Tx$, this direction is the residual $r=b-Ax$.

= Steepest descent method
{synonym}

= Lipschitz gradient
{parent=Gradient descent}
{wiki=Lipschitz_continuity}

A differentiable function has a $\beta$-Lipschitz gradient when
$$
\|\nabla f(x)-\nabla f(y)\|\leq\beta\|x-y\|.
$$

= Exact line search for a positive-definite quadratic
{parent=Gradient descent}

For $f(x)=x^TAx/2-b^Tx$ with $A$ positive definite and residual $r=b-Ax$, exact line search along $r$ uses
$$
t=\frac{r^Tr}{r^TAr}.
$$
The objective error contracts by
$$
1-\frac{(r^Tr)^2}{(r^TA^{-1}r)(r^TAr)}
\leq 1-\frac{\lambda_{\min}(A)}{\lambda_{\max}(A)}.
$$

= Conjugate gradient method
{c}
{parent=Gradient descent}
{wiki}

For a <positive-definite matrix> $A$, the conjugate gradient method chooses mutually $A$-conjugate search directions. Starting with $r_0=b-Ax_0$ and $p_0=r_0$, it uses
$$
\alpha_k=\frac{r_k^Tr_k}{p_k^TAp_k},\quad
x_{k+1}=x_k+\alpha_kp_k,\quad
r_{k+1}=r_k-\alpha_kAp_k,
$$
and $p_{k+1}=r_{k+1}+\beta_kp_k$ with
$\beta_k=r_{k+1}^Tr_{k+1}/(r_k^Tr_k)$.

= Krylov subspace
{title2=$\mathcal K_k(A,b)$}
{c}
{parent=Conjugate gradient method}
{wiki}

The order-$k$ Krylov subspace generated by $A$ and $b$ is
$$
\mathcal K_k(A,b)=\operatorname{span}\{b,Ab,\ldots,A^{k-1}b\}.
$$

= Finite termination of the conjugate gradient method
{parent=Conjugate gradient method}

In exact arithmetic, conjugate gradients reaches the exact solution in at most the number of distinct eigenvalues of $A$, and therefore in at most $n$ steps for an $n$-dimensional system.

= Heavy-ball method
{c}
{parent=Gradient descent}
{wiki=Momentum_(gradient_descent)#Heavy_ball}

The heavy-ball iteration adds momentum to gradient descent:
$$
x_{k+1}=x_k-\alpha\nabla F(x_k)+\beta(x_k-x_{k-1}).
$$

= Heavy-ball residual recurrence
{parent=Heavy-ball method}

For $F(x)=x^TAx/2-b^Tx$ and $r_k=b-Ax_k$,
$$
r_{k+1}=((1+\beta)I-\alpha A)r_k-\beta r_{k-1}.
$$
Thus $r_k$ is a polynomial in $A$ applied to the initial residual and lies in the corresponding <Krylov subspace>.

= Heavy-ball error propagation matrix
{title2=$M$}
{parent=Heavy-ball method}

For $e_k=x^*-x_k$,
$$
\binom{e_{k+1}}{e_k}
=
\begin{pmatrix}
(1+\beta)I-\alpha A&-\beta I\\
I&0
\end{pmatrix}
\binom{e_k}{e_{k-1}}.
$$
If $A$ is diagonal with entries $\lambda_i$, a coordinate permutation turns this matrix into blocks
$$
\begin{pmatrix}1+\beta-\alpha\lambda_i&-\beta\\1&0\end{pmatrix}.
$$

= Heavy-ball rate for a two-eigenvalue diagonal quadratic
{parent=Heavy-ball error propagation matrix}

For $A=\operatorname{diag}(1,\gamma)$, $\alpha=1/\gamma$, and $\beta=(1-\gamma^{-1/2})^2$, every error-propagation block has spectral radius at most $1-\gamma^{-1/2}$, and equality occurs.

= Peano kernel theorem
{parent=Numerical analysis}
{c}
{wiki}

The Peano kernel theorem represents a linear approximation error $L$ that annihilates polynomials below degree $r$ as
$$
L(f)=\int_a^bK(t)f^{(r)}(t)\,dt,
\qquad
K(t)=L\left(\frac{(x-t)_+^{r-1}}{(r-1)!}\right).
$$

= Four-point one-sided second-derivative formula
{parent=Peano kernel theorem}

At unit spacing, exactness through cubic polynomials gives
$$
f''(-1)\approx2f(-1)-5f(0)+4f(1)-f(2).
$$

= Sharp Peano-kernel constant for the four-point endpoint second derivative
{parent=Four-point one-sided second-derivative formula}

If the fourth-order Peano kernel of the four-point endpoint formula is nonnegative, its sharp sup-norm error constant is its integral. Evaluating the error on $(x+1)^4/24$ gives
$$
c=\int_{-1}^2K(t)\,dt=\frac{11}{12}.
$$

= Periodic trapezoidal Fourier aliasing
{parent=Numerical analysis}

For a two-periodic Fourier series sampled at $2N$ equally spaced points, discrete averaging retains exactly the modes divisible by $2N$:
$$
I_N(h)-I(h)=\sum_{j\ne0}\widehat h_{2Nj}.
$$
If $|\widehat h_n|\leq Mc^{|n|}$ with $0<c<1$, then
$$
|I_N-I|\leq\frac{2Mc^{2N}}{1-c^{2N}},
$$
which is exponentially small.

= Fourier-Galerkin matrix for a drift-diffusion equation
{c}
{parent=Numerical analysis}

For $u_t=u_{xx}-w'u_x$ and Fourier truncation $|n|\leq D$,
$$
\dot{\widehat u}_n=\sum_{|m|\leq D}B_{nm}\widehat u_m,
$$
where
$$
B_{nm}=-\pi^2n^2\delta_{nm}
+\pi^2(n-m)m\,\widehat w_{n-m}.
$$
For $w(x)=\cos\pi x$, the nonconstant block has Gershgorin discs in the closed left half-plane, while the constant mode lies in the kernel. Hence every eigenvalue has nonpositive real part and the matrix is singular.

= Fourier spectral method for variable-coefficient advection
{parent=Numerical analysis}

For $u_t+c(x)u_x=0$ with two-periodic Fourier truncations $|n|\leq d$, let $C_{nm}=\widehat c_{n-m}$ and $D_{mm}=m$. Fourier projection gives
$$
\dot{\widehat{\mathbf u}}=-i\pi CD\widehat{\mathbf u}.
$$

= Positive Fourier-symbol Toeplitz matrix
{parent=Fourier spectral method for variable-coefficient advection}

If a real Fourier polynomial $c(x)$ is strictly positive, then $C_{nm}=\widehat c_{n-m}$ is Hermitian positive definite because
$$
z^*Cz=\frac12\int_{-1}^1c(x)
\left|\sum_{m=-d}^dz_me^{i\pi mx}\right|^2dx>0
$$
for every nonzero $z$.

= Real spectrum of a positive-Hermitian times Hermitian product
{parent=Fourier spectral method for variable-coefficient advection}

If $C$ is Hermitian positive definite and $D$ is Hermitian, every eigenvalue of $CD$ is real. Equivalently, $CD$ is similar to the Hermitian matrix $C^{1/2}DC^{1/2}$.

= Explicit Euler instability on a nonzero imaginary eigenvalue
{parent=Fourier spectral method for variable-coefficient advection}

For $y'=i\omega y$ with real nonzero $\omega$, explicit Euler has amplification factor $1+i\omega h$, whose modulus $\sqrt{1+\omega^2h^2}$ exceeds one for every $h>0$.

= Runge-Kutta method
{parent=Numerical analysis}
{c}
{wiki}

= Local truncation error
{parent=Runge-Kutta method}

The local truncation error of a one-step method is the defect after one numerical step started from the exact solution:
$$
\tau_{n+1}
=y(t_{n+1})-y(t_n)-h\phi(t_n,y(t_n),h).
$$
A local error $O(h^{p+1})$ generally leads to global error $O(h^p)$ under a uniform stability bound.

= Implicit Runge-Kutta method
{parent=Runge-Kutta method}
{wiki}

An implicit Runge-Kutta method defines one or more stages through equations involving those same stages.

= Trapezoidal rule
{parent=Implicit Runge-Kutta method}
{wiki}

The trapezoidal ODE rule averages vector fields at the old and new states and is A-stable.

= Order of a Runge-Kutta method
{parent=Runge-Kutta method}
{wiki}

The order is the highest power through which the one-step expansion matches the exact Taylor expansion.

= Simplex algorithm
{parent=Numerical analysis}
{wiki}

= Sparse optimization
{parent=Numerical analysis}
{wiki}

= Stability function
{parent=Numerical analysis}
{wiki}

= Multistep method
{parent=Numerical analysis}
{wiki=Linear_multistep_method}

A linear multistep method approximates an ODE using several previous solution and derivative values.

= Zero-stability
{parent=Multistep method}
{wiki}

Zero-stability requires the roots of the first characteristic polynomial to lie in the closed unit disc, with unit-modulus roots simple.

= Root condition for a multistep method
{parent=Zero-stability}

The root condition is the polynomial criterion equivalent to zero-stability for a linear multistep method.

= Dahlquist equivalence theorem
{parent=Multistep method}
{c}
{wiki}

A consistent linear multistep method is convergent exactly when it is zero-stable.

= Numerical linear algebra
{parent=Numerical analysis}
{wiki}

Numerical linear algebra develops stable finite algorithms for matrix computations.

= Gaussian elimination
{parent=Numerical linear algebra}
{c}
{wiki}

Gaussian elimination applies elementary row operations to reduce a linear system to triangular form. Dense elimination uses $O(n^3)$ arithmetic operations.

= LDL decomposition
{title2=$A=LDL^T$}
{parent=Numerical linear algebra}
{c}
{wiki=Cholesky_decomposition\#LDL_decomposition}

An LDL decomposition of a real symmetric matrix is
$$
A=LDL^T,
$$
where $L$ is unit lower triangular and $D$ is diagonal. Symmetric elimination computes it one <Schur complement> at a time. The matrix is positive definite exactly when every diagonal pivot in $D$ is positive.

= Machine precision
{title2=$\epsilon_{\rm mach}$}
{parent=Numerical linear algebra}
{wiki=Machine_epsilon}

Machine precision is the characteristic relative rounding scale of a floating-point system.

= Eigenpair residual
{title2=$r=Av-\lambda v$}
{parent=Numerical linear algebra}

For an approximate eigenpair $(\widetilde\lambda,\widetilde v)$, the residual is
$$
r=A\widetilde v-\widetilde\lambda\widetilde v.
$$

= Backward error of an approximate eigenpair
{parent=Eigenpair residual}

If $\|\widetilde v\|_2=1$, the smallest operator-norm perturbation that makes $(\widetilde\lambda,\widetilde v)$ an exact eigenpair has norm $\|r\|_2$. One such rank-one perturbation is
$$
E=-r\widetilde v^T.
$$

= Householder QR decomposition
{parent=Numerical linear algebra}
{c}
{wiki}

Successive Householder reflections zero subdiagonal column entries and factor a matrix into an orthogonal factor and an upper-triangular factor.

= Householder tridiagonalization
{parent=Numerical linear algebra}
{c}
{wiki}

Orthogonal Householder similarities reduce a real symmetric matrix to symmetric tridiagonal form.

= Unshifted QR algorithm
{parent=Numerical linear algebra}
{wiki=QR_algorithm}

The unshifted QR algorithm factors $A_k=Q_kR_k$ and forms $A_{k+1}=R_kQ_k=Q_k^TA_kQ_k$, preserving eigenvalues while driving the matrix toward block diagonal form.

= Accumulated QR factorization identity
{parent=Unshifted QR algorithm}
{c}

Let
$$
\overline Q_k=Q_0Q_1\cdots Q_k,
\qquad
\overline R_k=R_kR_{k-1}\cdots R_0.
$$
Then
$$
A_{k+1}=\overline Q_k^TA\overline Q_k,
\qquad
A^{k+1}=\overline Q_k\overline R_k.
$$
Thus $\overline Q_k\overline R_k$ is a <QR decomposition> of the matrix power $A^{k+1}$. In particular, the first $r$ columns of $\overline Q_k$ span the same subspace as the first $r$ columns of $A^{k+1}$ whenever those columns are independent.

= Symmetric bandwidth preservation under QR iteration
{parent=Unshifted QR algorithm}

An unshifted QR step preserves the bandwidth of a real symmetric banded matrix. A Givens-rotation implementation exposes this as bulges created during triangularization and removed when the factors are multiplied in reverse order.

= Simultaneous iteration interpretation of the QR algorithm
{parent=Unshifted QR algorithm}

The accumulated orthogonal factor in QR iteration is the $Q$ factor of a matrix power, so its leading columns span the same spaces as simultaneous power iteration.

= Two-column dominant-subspace condition
{parent=Simultaneous iteration interpretation of the QR algorithm}

Let a real <symmetric matrix> have an <orthonormal eigenbasis> $w_1,\ldots,w_n$, with
$$
|\lambda_1|\leq\cdots\leq|\lambda_{n-2}|<
|\lambda_{n-1}|=|\lambda_n|.
$$
Write two starting vectors as
$$
u=\sum_i b_iw_i,
\qquad
v=\sum_i c_iw_i.
$$
For two-column <subspace iteration> to recover
$\operatorname{span}(w_{n-1},w_n)$, the two projections onto that dominant subspace must be independent. The exact condition is
$$
\det\begin{pmatrix}
b_{n-1}&c_{n-1}\\
b_n&c_n
\end{pmatrix}
=b_{n-1}c_n-b_nc_{n-1}\ne0.
$$
Requiring each of the four coefficients to be nonzero does not imply this determinant condition.

= Block deflation in the QR algorithm
{parent=Unshifted QR algorithm}

When an off-block subdiagonal entry tends to zero, QR iteration asymptotically separates the corresponding invariant spectral blocks even if eigenvalues within one block have equal modulus.

= Stationary iterative method for a linear system
{parent=Numerical linear algebra}
{wiki=Iterative_method}

A stationary iteration has the form
$$
x^{(k+1)}=Hx^{(k)}+v,
$$
with a fixed iteration matrix $H$. It converges for every initial vector exactly when the <spectral radius> satisfies $\rho(H)<1$.

= Matrix splitting
{parent=Stationary iterative method for a linear system}
{wiki=Matrix_splitting}

A splitting $A=M-N$ with invertible $M$ produces
$$
x^{(k+1)}=M^{-1}Nx^{(k)}+M^{-1}b.
$$

= Householder-John theorem
{parent=Matrix splitting}
{c}

Let $A=M-N$ be Hermitian positive definite. If $M^*+N$ is also Hermitian positive definite, then $M$ is invertible and
$$
\rho(M^{-1}N)<1.
$$
Thus the stationary iteration associated with the splitting converges from every initial vector.

= Jacobi method
{parent=Stationary iterative method for a linear system}
{c}
{wiki=Jacobi_method}

Writing $A=D+L+U$, where $D$ is diagonal and $L,U$ are strictly triangular, the Jacobi iteration is
$$
x^{(k+1)}=D^{-1}\left(b-(L+U)x^{(k)}\right).
$$

= Jacobi convergence for a symmetric positive-definite tridiagonal matrix
{parent=Jacobi method}

If $A=D+L+L^T$ is symmetric positive definite and tridiagonal, set $S=\operatorname{diag}(1,-1,1,-1,\ldots)$. Then
$$
SAS=D-L-L^T
$$
is positive definite. Applying the <Householder-John theorem> to $M=D$ and $N=-(L+L^T)$ proves that the Jacobi iteration converges.

= Linear stability domain
{parent=Numerical analysis}
{wiki}

For a one-step method applied to $y'=\lambda y$, write $y_{n+1}=R(h\lambda)y_n$. Its linear stability domain is
$$
\mathcal S=\{z\in\mathbb C:|R(z)|\leq1\}.
$$
Forward Euler has $R(z)=1+z$, while backward Euler has $R(z)=(1-z)^{-1}$ and is A-stable.

= Absolute stability
{synonym}

= Explicit Euler method
{parent=Linear stability domain}
{wiki=Euler_method}

Explicit Euler advances $y'=f(t,y)$ by $y_{n+1}=y_n+h f(t_n,y_n)$. Its <stability function> is $R(z)=1+z$.

= Forward Euler method
{synonym}

= A-stability
{parent=Linear stability domain}
{c}
{wiki}

A method is A-stable when its stability domain contains the closed left half-plane.

= A-stable
{c}
{synonym}

= A-stability of a symmetric two-stage implicit Runge-Kutta family
{parent=A-stability}
{c}

For the two-stage <implicit Runge-Kutta method> with
$$
A=
\begin{pmatrix}
\frac14&\frac14-a\\
\frac14+a&\frac14
\end{pmatrix},
\qquad
b=\begin{pmatrix}\frac12\\\frac12\end{pmatrix},
$$
the <stability function> is
$$
R(z)=\frac{2+z+2a^2z^2}{2-z+2a^2z^2}.
$$
For $z=x+iy$,
$$
|2-z+2a^2z^2|^2-|2+z+2a^2z^2|^2
=-8x(1+a^2|z|^2).
$$
The denominator has no zero in the closed left half-plane, so the method is <A-stable> for every real $a$.

= Stiff two-mode linear system
{parent=Linear stability domain}

If a linear system has negative eigenvalues with widely separated magnitudes, an explicit method can be forced to resolve the fastest decaying mode solely for stability. For eigenvalues $-1$ and $-100$, forward Euler requires $h\leq0.02$, whereas backward Euler is stable for every positive step size.

= Milne device for forward and backward Euler
{parent=Linear stability domain}
{c}

Starting at the same value, forward and backward Euler have opposite leading local errors. Their half-difference therefore estimates either local error:
$$
E_n=\frac12\lVert y_B-y_F\rVert.
$$
For a first-order method, a local-tolerance controller consequently scales the next step by $(\mathrm{tol}/E_n)^{1/2}$, usually with a safety factor.

= Orthogonal polynomial
{parent=Numerical analysis}
{wiki}

Orthogonal polynomials of distinct degrees are orthogonal under a positive weighted inner product.

= Orthogonal polynomials
{synonym}

= Monic orthogonal polynomial
{parent=Orthogonal polynomial}

A monic orthogonal polynomial has leading coefficient one. For a fixed positive weight, there is a unique monic orthogonal polynomial of each degree.

= Monic orthogonal polynomials
{synonym}

= Hermite polynomial
{parent=Orthogonal polynomial}
{c}
{wiki}

= Zeros of orthogonal polynomials
{parent=Orthogonal polynomial}
{wiki}

A degree-n orthogonal polynomial for a positive interval weight has n simple zeros in the interval interior.

= Jacobi matrix
{parent=Orthogonal polynomial}
{c}
{wiki}

A Jacobi matrix is a symmetric tridiagonal matrix whose characteristic polynomials obey the orthogonal-polynomial three-term recurrence.

= Quadrature rule
{parent=Numerical analysis}
{wiki=Numerical_integration}

A quadrature rule approximates a weighted integral by a finite weighted sum of function values.

= Nodal-polynomial criterion for quadrature exactness
{parent=Quadrature rule}

An $(n+1)$-node <quadrature rule> already exact through degree $n$ is exact through degree $n+1+k$ exactly when its nodal polynomial
$$
Q_{n+1}(x)=\prod_{i=0}^n(x-x_i)
$$
is orthogonal to every polynomial of degree at most $k$.

= Degree ceiling for quadrature exactness
{parent=Quadrature rule}

No $(n+1)$-node quadrature rule for a positive interval weight can be exact through degree $2n+2$, because it evaluates the nonnegative nodal square $Q_{n+1}^2$ as zero although its integral is positive.

= Positivity of quadrature weights from degree 2n exactness
{parent=Quadrature rule}

If an $(n+1)$-node quadrature rule for a positive interval weight is exact through degree $2n$, every weight is positive. Indeed, applying the rule to the square of the corresponding degree-$n$ Lagrange cardinal polynomial isolates that weight.

= Convergence of positive quadrature rules
{parent=Quadrature rule}

If each $(n+1)$-node quadrature rule is exact through degree $n$ and has positive weights, then it converges on every continuous function. The <Weierstrass approximation theorem> reduces the error to the uniform polynomial-approximation error, while exactness on constants controls the sum of the weights.

= Moment matching
{parent=Quadrature rule}

A quadrature rule is exact through degree $d$ precisely when its node weights reproduce the first $d+1$ moments of the integration measure.

= Asymptotic expansion
{parent=Analysis}
{wiki}

= Asymptotic sequence
{parent=Asymptotic expansion}

Functions $(\phi_n)$ form an asymptotic sequence at $x_0$ when $\phi_{n+1}=o(\phi_n)$. The notation
$$
f\sim\sum_{n\geq0}a_n\phi_n
$$
means that for every $N\geq0$,
$$
f-\sum_{n=0}^Na_n\phi_n=o(\phi_N).
$$
Successive division of the remainder by $\phi_N$ uniquely recovers every coefficient.

= Harmonic-mean refinement of an asymptotic sequence
{parent=Asymptotic sequence}

For a positive asymptotic sequence $(\phi_n)$, define $\psi_0=\phi_0$ and
$$
\psi_n=\frac{\phi_{n-1}\phi_n}{\phi_{n-1}+\phi_n}.
$$
Then $\psi_n\sim\phi_n$ term by term and $(\psi_n)$ is again an asymptotic sequence.

= Geometric-mean refinement of an asymptotic sequence
{parent=Asymptotic sequence}

For $\chi_0=\phi_0$ and $\chi_n=\sqrt{\phi_{n-1}\phi_n}$, the sequence $(\chi_n)$ is asymptotic, but $\chi_n/\phi_n\to\infty$ for every $n\geq1$.

= Termwise equivalent asymptotic scales need not preserve expansions
{parent=Asymptotic sequence}

Even if $\phi_n\sim\psi_n$ for every $n$, an expansion in the scale $(\phi_n)$ need not induce one in $(\psi_n)$. A perturbation of $\psi_0$ that is smaller than $\phi_0$ but larger than $\phi_1$ supplies a counterexample.

= Laplace's method
{c}
{parent=Asymptotic expansion}
{wiki=Laplace%27s_method}

If a smooth real phase $\phi$ has a unique nondegenerate interior maximum at $t_0$, then
$$
\int a(t)e^{x\phi(t)}dt
\sim a(t_0)e^{x\phi(t_0)}
\sqrt{\frac{2\pi}{x|\phi''(t_0)|}}.
$$
For a maximum at an endpoint $b$ with $\phi'(b)>0$, the leading factor is $a(b)e^{x\phi(b)}/(x\phi'(b))$.

= Laplace asymptotic for the derivative of the Gamma function
{parent=Laplace's method}

After $t=zs$, the phase in the integral for $\Gamma'(z)$ becomes $\log s-s$, with maximum $-1$ and second derivative $-1$ at $s=1$. Hence
$$
\Gamma'(z)\sim\sqrt{\frac{2\pi}{z}}e^{z\log z-z}\log z.
$$

= Method of steepest descent
{parent=Asymptotic expansion}
{wiki}

A contour integral with a large exponential parameter is deformed through saddle points along curves on which the real part of the phase decreases most rapidly and the imaginary part is locally constant.

= Saddle point
{parent=Method of steepest descent}
{wiki=Saddle_point}

For a holomorphic phase $\phi$, a saddle point is a critical point $z_0$ with $\phi'(z_0)=0$. At a simple saddle, $\phi''(z_0)\ne0$, and the quadratic Taylor term determines the local descent directions.

= Simple-saddle contribution in steepest descent
{parent=Method of steepest descent}

If $\phi'(z_0)=0$, $\phi''(z_0)\ne0$, and the oriented steepest-descent tangent has angle $\alpha$, then
$$
\int_C f(z)e^{x\phi(z)}\,dz
\sim
f(z_0)e^{x\phi(z_0)+i\alpha}
\sqrt{\frac{2\pi}{x|\phi''(z_0)|}}.
$$
The tangent condition is
$$
\arg\phi''(z_0)+2\alpha=\pi\pmod{2\pi}.
$$

= Bessel function
{parent=Analysis}
{c}
{wiki}

= Modified Bessel function
{parent=Bessel function}
{wiki=Bessel_function\#Modified_Bessel_functions}

The modified Bessel equation of order zero is
$$
z^2y''+zy'-z^2y=0.
$$
Its solution regular at zero and normalized by $I_0(0)=1$ is the modified Bessel function $I_0$.

= Laplace integral method for a differential equation
{parent=Modified Bessel function}
{c}

Laplace's integral method seeks $y(z)=\int_Ce^{zt}f(t)\,dt$. Substitution into the differential equation and integration by parts transfer multiplication by $z$ into differentiation with respect to $t$, producing a first-order equation for $f$ and endpoint conditions for $C$.

= Laplace integral solution of a singular third-order equation
{parent=Laplace integral method for a differential equation}

For $x>0$, substituting $y(x)=\int_\gamma e^{xt}f(t)\,dt$ into $xy^{(3)}+2y=0$ and integrating by parts gives
$$
[e^{xt}t^3f(t)]_{\partial\gamma}
+\int_\gamma e^{xt}\{2f-(t^3f)'\}\,dt=0.
$$
The amplitude equation $(t^3f)'=2f$ has solution $f(t)=Ct^{-3}e^{-1/t^2}$. On $\gamma=(-\infty,0)$ both endpoint terms vanish, and hence
$$
y(x)=C\int_{-\infty}^0e^{xt-t^{-2}}t^{-3}\,dt
=C_1\int_0^\infty u e^{-u^2-x/u}\,du.
$$

= Laguerre contour-integral amplitude
{parent=Laplace integral method for a differential equation}
{c}

For the Laguerre equation
$$
zy''+(1-z)y'+\lambda y=0,
$$
the ansatz $y=\int_\gamma e^{zt}f(t)\,dt$ gives
$$
f(t)=t^{-\lambda-1}(t-1)^\lambda.
$$
The contour must make the endpoint contribution
$$
\left[e^{zt}t^{-\lambda}(t-1)^{\lambda+1}\right]_{\partial\gamma}
$$
vanish.

= Finite Laguerre contour solution
{parent=Laguerre contour-integral amplitude}
{c}

A finite contour based at an integrable branch point, or a closed contour around an integer-order pole, gives a solution analytic at $z=0$. It is therefore a constant multiple of the unique Frobenius power-series solution there.

= Integer-parameter Laguerre contour residues
{parent=Finite Laguerre contour solution}
{c}

For $\lambda=N\geq0$, the amplitude $(t-1)^N/t^{N+1}$ has a pole at zero whose contour residue is a degree-$N$ Laguerre polynomial in $z$. For $\lambda=-m<0$, the amplitude $t^{m-1}/(t-1)^m$ has a pole at one whose residue is $e^z$ times a polynomial of degree $m-1$.

= Real integral representation of the modified Bessel function I0
{parent=Laplace integral method for a differential equation}

The normalized order-zero function has the representation
$$
I_0(z)=\frac1\pi\int_{-1}^1\frac{e^{zs}}{\sqrt{1-s^2}}\,ds.
$$

= Branch cut
{parent=Analysis}
{wiki}

= Monodromy of a meromorphic integral
{parent=Branch cut}

Analytic continuation of an integral of a meromorphic function around a closed loop changes its value by $2\pi i$ times the winding-number-weighted sum of enclosed residues.

= Residue contribution to path dependence
{parent=Monodromy of a meromorphic integral}

Two integration paths with the same endpoints differ by a closed-path period determined by the residues and winding numbers of their concatenation.

= Branch cut joining cancelling poles
{parent=Branch cut}

If a cut joins two poles with opposite residues, every loop avoiding the cut has equal winding number about the endpoints, so the two residue contributions cancel.

= Path independence by vanishing periods
{parent=Branch cut}

A holomorphic or meromorphic integrand has a single-valued path integral on a domain whenever its integral around every closed curve in that domain vanishes.

= Cauchy integral formula
{parent=Analysis}
{c}
{wiki}

= Cauchy estimate
{parent=Cauchy integral formula}
{c}
{wiki=Cauchy%27s_integral_formula#Consequences}

If $f$ is holomorphic on a neighbourhood of the closed disc $|z-a|\leq R$ and $|f(z)|\leq M$ on its boundary, differentiating the <Cauchy integral formula> gives
$$
|f^{(n)}(a)|\leq \frac{n!M}{R^n}.
$$

= Cauchy-Riemann equations
{parent=Analysis}
{c}
{wiki}

= Complex logarithm
{parent=Analysis}
{wiki}

= Complex exponentiation
{parent=Complex logarithm}
{wiki=Exponentiation#Complex_exponents}

For nonzero complex $z$ and complex $w$, the multivalued power is
$$
z^w=\left\{\exp\!\left(w(\log|z|+i(\arg z+2\pi k))\right):k\in\mathbb Z\right\}.
$$

= Analytic logarithm on the positive-axis slit plane
{parent=Complex logarithm}

On $\mathbb C\setminus[0,\infty)$, choose the argument in $(0,2\pi)$. Then
$$
L(z)=\log|z|+i\arg z
$$
is an analytic logarithm. Locally this follows by scaling the convergent series
$$
-\sum_{n=1}^{\infty}\frac{(1-z)^n}{n},
\qquad |z-1|<1.
$$

= Positive-axis keyhole beta integral
{parent=Analytic logarithm on the positive-axis slit plane}

For $-1<\alpha<1$,
$$
\int_0^\infty\frac{x^\alpha}{(1+x)^2}\,dx
=\frac{\pi\alpha}{\sin(\pi\alpha)},
$$
with value one at $\alpha=0$ by continuity. A keyhole contour around the positive axis gives a jump factor $1-e^{2\pi i\alpha}$, while the double pole at $-1$ has residue $-\alpha e^{\pi i\alpha}$.

= Critical point
{parent=Analysis}
{wiki}

= Critical set
{parent=Critical point}

The critical set of a differentiable map consists of the points where its derivative fails to have full rank.

= Local minimum
{parent=Critical point}
{wiki=Maxima_and_minima}

A function has a local minimum at $x_0$ when $f(x)\geq f(x_0)$ throughout some neighbourhood of $x_0$. For a twice differentiable function, its <Hessian matrix> at an interior local minimum is <positive semidefinite matrix>[positive semidefinite].

= Damped harmonic oscillator
{parent=Analysis}
{wiki}

= Forced harmonic oscillator
{parent=Damped harmonic oscillator}
{wiki}

A forced harmonic oscillator responds at both its natural and forcing frequencies.

= Differentiable function
{parent=Analysis}
{wiki}

= Differentiable
{synonym}

= Smooth function
{parent=Differentiable function}
{wiki=Smoothness}

A smooth function has continuous derivatives of every order.

= Existence of partial derivatives does not imply their continuity
{parent=Differentiable function}

A function can be differentiable even when its partial derivatives are discontinuous. For example,
$$
f(x,y)=(x^2+y^2)\sin\frac1{\sqrt{x^2+y^2}}
$$
away from the origin, with value zero at the origin, is differentiable there while its partial derivatives oscillate along the coordinate axes.

= Differentiation under the integral sign
{parent=Analysis}
{wiki}

If $g(t,x)$ and its partial derivative $D_i g(t,x)$ are continuous for $t$ in a compact interval and $x$ in an open set, then
$$
D_i\int_a^b g(t,x)\,dt
=\int_a^b D_i g(t,x)\,dt.
$$
Apply the <fundamental theorem of calculus> to the difference quotient in the $i$th coordinate and use uniform continuity on a compact neighbourhood.

= Second-order Hadamard lemma
{parent=Differentiation under the integral sign}
{c}
{wiki=Hadamard%27s_lemma}

For a <smooth function> $f(x,y)$,
$$
f(x,y)=f(x,0)+yD_2f(x,0)+y^2h(x,y),
$$
where the smooth remainder is
$$
h(x,y)=\int_0^1(1-t)D_2^2f(x,ty)\,dt.
$$

= Dirichlet problem
{parent=Analysis}
{c}
{wiki}

= Euler-Lagrange equation
{parent=Analysis}
{c}
{wiki}

For $I[y]=\int_a^bF(x,y,y')\,dx$, stationarity under fixed-endpoint variations gives
$$
\frac d{dx}F_{y'}-F_y=0.
$$

= Beltrami identity
{parent=Euler-Lagrange equation}
{c}
{wiki}

When a variational integrand $f(z,z')$ has no explicit dependence on the independent variable, every extremal satisfies
$$
f-z'f_{z'}=\text{constant}.
$$

= Catenary
{parent=Euler-Lagrange equation}
{wiki}

A catenary is a curve of the form $y=a\cosh((x-x_0)/a)+y_0$. It is the equilibrium shape of a uniform flexible chain in a uniform gravitational field.

= Natural boundary conditions for a free endpoint
{parent=Euler-Lagrange equation}

When the values of $y$ are free at both fixed endpoint locations, integration by parts leaves the boundary term $[F_{y'}\eta]_a^b$. Since the endpoint variations are arbitrary, stationarity requires
$$
F_{y'}(a)=F_{y'}(b)=0.
$$

= Free-endpoint normal mode of a coupled variational functional
{parent=Natural boundary conditions for a free endpoint}

For
$$
I[y,z]=\int_0^{x_0}(y'^2+z'^2+2yz)\,dx,
\qquad y(0)=z(0)=0,
$$
with free terminal values, the Euler-Lagrange equations and natural conditions give a nonzero stationary family exactly when $x_0=(k+\tfrac12)\pi$. It is
$$
y=C\sin x,
\qquad z=-C\sin x.
$$

= Euler-Lagrange equations for two fields
{parent=Euler-Lagrange equation}
{c}

For
$$
\mathcal L[u,v]=\iint
f(x,y,u,v,u_x,u_y,v_x,v_y)\,dx\,dy,
$$
independent variations and integration by parts give
$$
f_u-\partial_xf_{u_x}-\partial_yf_{u_y}=0,
\qquad
f_v-\partial_xf_{v_x}-\partial_yf_{v_y}=0.
$$

= Exponential decay
{parent=Analysis}
{wiki}

= First-step analysis
{parent=Analysis}
{wiki}

First-step analysis conditions on the first transition of a stochastic process, turning hitting probabilities, hitting times, and accumulated rewards into linear recurrence equations.

= Forced oscillator
{parent=Analysis}
{wiki}

= Steady-state response
{parent=Forced oscillator}
{wiki=Steady_state}

For a damped linear oscillator driven periodically, the steady-state response is the periodic particular solution left after homogeneous transients decay.

= Fourier series
{parent=Analysis}
{c}
{wiki}

= Leibniz formula for pi
{parent=Fourier series}
{c}
{wiki=Leibniz_formula_for_%CF%80}

The Leibniz formula is the conditionally convergent alternating series
$$
\frac\pi4=\sum_{r=0}^{\infty}\frac{(-1)^r}{2r+1}.
$$

= Trigonometric polynomial
{parent=Fourier series}
{wiki}

A trigonometric polynomial is a finite linear combination of complex exponentials $e^{inx}$, or equivalently of sines and cosines. If it vanishes on an interval, all its coefficients vanish.

= Fourier coefficient
{title2=$\widehat f_n$}
{parent=Fourier series}
{wiki=Fourier_series#Exponential_form}

For a function of period $L$, its complex Fourier coefficients are
$$
\widehat f_n=\frac1L\int_{x_0}^{x_0+L}f(x)e^{-2\pi inx/L}\,dx.
$$

= Fourier cosine series
{parent=Fourier series}
{c}
{wiki=Fourier_series}

A Fourier cosine series has the form
$$
\frac{a_0}{2}+\sum_{n=1}^{\infty}a_n\cos(nx).
$$
It represents the <even function> obtained by reflecting its data across the origin.

= Fourier series of x cubed minus pi squared x
{parent=Fourier series}
{c}

On $(-\pi,\pi)$,
$$
x^3-\pi^2x
=12\sum_{n=1}^{\infty}\frac{(-1)^n}{n^3}\sin(nx).
$$
Parseval's identity and direct integration of the square give
$$
\sum_{n=1}^{\infty}\frac1{n^6}=\frac{\pi^6}{945}.
$$

= Fourier transform
{title2=$\mathcal F$}
{parent=Analysis}
{c}
{wiki}

For $f\in L^1(\mathbb R^n)$, the angular-frequency Fourier transform is
$$
\widehat f(\xi)
=\int_{\mathbb R^n}f(x)e^{-ix\cdot\xi}\,dx.
$$

= Scaling property of the Fourier transform
{parent=Fourier transform}

If $f_\lambda(x)=f(\lambda x)$ on $\mathbb R^n$ with $\lambda>0$, then
$$
\widehat {f_\lambda}(\xi)
=\lambda^{-n}\widehat f(\xi/\lambda),
$$
and consequently
$$
\|f_\lambda\|_p=\lambda^{-n/p}\|f\|_p,
\qquad
\|\widehat {f_\lambda}\|_q
=\lambda^{-n(1-1/q)}\|\widehat f\|_q.
$$

= Scaling necessity for an Lp to Lq Fourier bound
{parent=Scaling property of the Fourier transform}

If a nontrivial uniform estimate
$$
\|\widehat f\|_q\leq C\|f\|_p
$$
holds for all integrable $L^p$ functions on $\mathbb R^n$, dilation by every $\lambda>0$ forces
$$
1-\frac1q=\frac1p.
$$
Thus $q$ must be the <conjugate exponents>[conjugate exponent] of $p$.

= Fourier transform of inverse distance in three dimensions
{title2=$\mathcal F(|x|^{-1})$}
{parent=Fourier transform}

As a <tempered distribution> on $\mathbb R^3$,
$$
\mathcal F(|x|^{-1})(\xi)=C|\xi|^{-2}
$$
for a nonzero convention-dependent constant $C$. One proof writes
$$
|x|^{-1}=C_1\int_0^\infty t^{-1/2}e^{-t|x|^2}\,dt
$$
and applies the <Fourier transform of a Gaussian> before substituting $u=|\xi|^2/(4t)$.

= Hilbert transform
{parent=Fourier transform}
{c}
{wiki}

With one common sign convention,
$$
Hf(y)=\frac1\pi\operatorname{PV}
\int_{-\infty}^{\infty}\frac{f(x)}{y-x}\,dx.
$$
Reversing the denominator reverses the sign.

= Sinc-squared integral
{parent=Fourier transform}
{wiki}

With the standard Fourier convention, the integral of sinc squared over the real line equals pi.

= Gradient flow
{parent=Analysis}
{wiki}

= Green function
{parent=Analysis}
{c}
{wiki}

= One-dimensional modified Helmholtz Green function
{parent=Green function}

For $\operatorname{Re}m>0$, the decaying solution of
$$
-G''+m^2G=\delta_0
$$
is
$$
G(x)=\frac{e^{-m|x|}}{2m}.
$$
It is continuous at zero and has derivative jump $G'(0+)-G'(0-)=-1$.

= Dirichlet half-line Green function for d2 minus 1
{parent=One-dimensional modified Helmholtz Green function}

For $x,\xi>0$, the <Green function> satisfying
$$
(\partial_x^2-1)G(x;\xi)=\delta(x-\xi),\qquad
G(0;\xi)=0,\qquad G(x;\xi)\to0
$$
as $x\to\infty$ is
$$
G(x;\xi)=-\sinh(\min\{x,\xi\})e^{-\max\{x,\xi\}}.
$$

= Neumann Green function for a second-order ordinary differential equation
{parent=Green function}
{c}

Let $y_1,y_2$ solve $y''+\alpha y'+\beta y=0$ with $y_1'(0)=0$ and $y_2'(1)=0$. For Neumann boundary conditions, the Green function is
$$
G(x,\xi)=\frac1{W(\xi)}
\begin{cases}
y_1(x)y_2(\xi),&x<\xi,\\
y_2(x)y_1(\xi),&x>\xi,
\end{cases}
$$
where $W=y_1y_2'-y_1'y_2$. The derivative jump is one. If $\alpha=0$, the Abel identity makes $W$ constant and $G$ symmetric.

= Causal Green function
{parent=Green function}
{wiki}

A causal Green function vanishes before the source time and has the derivative jump required by a delta source.

= One-dimensional outgoing Green function
{parent=Green function}

For positive $k$, $e^{ik|x-x'|}$ is outgoing on both sides of $x'$ and obeys
$$
\left(\frac{d^2}{dx^2}+k^2\right)e^{ik|x-x'|}=2ik\delta(x-x').
$$

= Hermite differential equation
{parent=Analysis}
{c}
{wiki}

= Higher-order Euler-Lagrange equation
{parent=Analysis}
{wiki}

For the <functional>
$$
L[y]=\int_a^b F(x,y,y',\ldots,y^{(m)})\,dx,
$$
the vanishing of its <first variation> under fixed endpoint data gives
$$
\sum_{j=0}^m(-1)^j\frac{d^j}{dx^j}F_{y^{(j)}}=0.
$$

= Clamped-free beam under uniform load and endpoint force
{parent=Higher-order Euler-Lagrange equation}

For
$$
E[y]=\int_0^L\left(\frac A2(y'')^2+\rho gy\right)\,dx
$$
with the <boundary conditions> $y(0)=y'(0)=0$, $y''(L)=0$, and $-Ay'''(L)=F$, the <higher-order Euler-Lagrange equation> is
$$
Ay''''+\rho g=0.
$$
Its solution splits as
$$
y=y_0+y_F,
\qquad
y_0=-\frac{\rho g}{24A}x^2(6L^2-4Lx+x^2),
\qquad
y_F=\frac{F}{6A}x^2(3L-x).
$$

= Endpoint force derivative of clamped-free beam energy
{parent=Clamped-free beam under uniform load and endpoint force}

For the <clamped-free beam under uniform load and endpoint force>, the additional minimized internal energy is
$$
\mathcal E(F)=\frac{F^2L^3}{6A}.
$$
Its <derivative> is the endpoint <displacement> caused by the endpoint <force>:
$$
\frac{d\mathcal E}{dF}=\frac{FL^3}{3A}=h.
$$

= Laurent series
{parent=Analysis}
{c}
{wiki}

= Laurent theorem
{parent=Laurent series}
{c}
{wiki=Laurent_series}

An analytic function on an annulus $r<|z-a|<R$ has a unique locally uniformly convergent expansion
$$
f(z)=\sum_{n=-\infty}^{\infty}c_n(z-a)^n,
\qquad
c_n=\frac1{2\pi i}\oint_C\frac{f(\zeta)}{(\zeta-a)^{n+1}}\,d\zeta,
$$
where $C$ is any positively oriented circle around $a$ in the annulus.

= Method of characteristics
{parent=Analysis}
{wiki}

= Differential equation
{parent=Analysis}
{wiki}

= Ordinary differential equation
{parent=Differential equation}
{wiki}

= Linear differential equation
{parent=Ordinary differential equation}
{wiki}

A linear differential equation is linear in the unknown function and its derivatives.

= Inhomogeneous linear differential equation
{parent=Linear differential equation}

An inhomogeneous linear differential equation has a nonzero forcing term. Its general solution is one particular solution plus the general solution of the associated homogeneous equation.

= Local existence for a scalar autonomous ordinary differential equation with continuous vector field
{parent=Ordinary differential equation}

For continuous $\phi:\mathbb R\to\mathbb R$, the initial-value problem
$$
f'(t)=\phi(f(t)),
\qquad f(0)=0,
$$
has a local continuously differentiable solution. If $\phi(0)=0$, use the constant solution; otherwise locally invert $F(x)=\int_0^xdu/\phi(u)$.

= Singular perturbation
{parent=Ordinary differential equation}
{wiki}

A singular perturbation multiplies a highest derivative or otherwise changes the limiting equation's order when its small parameter is set to zero. It commonly separates fast and slow time scales.

= Lie point symmetry of an ordinary differential equation
{parent=Ordinary differential equation}
{wiki=Lie_point_symmetry}

A Lie point symmetry is a local one-parameter transformation of the independent and dependent variables that maps solution graphs of a differential equation to solution graphs.

= Prolongation of a vector field
{parent=Lie point symmetry of an ordinary differential equation}
{wiki=Jet_bundle#Prolongation}

For $V=\xi(x,u)\partial_x+\eta(x,u)\partial_u$, its prolongation to derivatives through order $n$ is
$$
\operatorname{pr}^{(n)}V
=V+\sum_{j=1}^n\eta^{(j)}\partial_{u^{(j)}},
\qquad
\eta^{(j)}=D_x\eta^{(j-1)}-u^{(j)}D_x\xi.
$$
It generates a symmetry of $\Delta=0$ exactly when $\operatorname{pr}^{(n)}V(\Delta)$ vanishes on the equation manifold.

= Total derivative operator
{title2=$D_x$}
{parent=Prolongation of a vector field}

On jet coordinates,
$$
D_x=\partial_x+u'\partial_u+u''\partial_{u'}+\cdots.
$$
It differentiates a differential function along prolonged solution graphs.

= Affine-scaling symmetry of u double prime equals u prime squared over u minus u squared
{title2=$u''=(u')^2/u-u^2$}
{parent=Prolongation of a vector field}

The vector fields $(cx+d)\partial_x-2cu\partial_u$ generate
$$
(x,u)\longmapsto(\lambda x+a,\lambda^{-2}u).
$$
Their second prolongations multiply the differential equation by $-4c$, so they are infinitesimal Lie symmetries.

= Boundary value problem
{parent=Ordinary differential equation}
{wiki}

A boundary value problem asks for a differential-equation solution satisfying conditions at more than one point or boundary component.

= Boundary condition
{parent=Ordinary differential equation}
{wiki}

A boundary condition prescribes values of an unknown <function> or its <derivatives> at the boundary of the domain of a <differential equation>.

= Dirichlet boundary condition
{parent=Boundary condition}
{wiki=Dirichlet_boundary_condition}

A Dirichlet boundary condition prescribes the value of the unknown function on the boundary.

= Neumann boundary condition
{parent=Boundary condition}
{wiki=Neumann_boundary_condition}

A Neumann boundary condition prescribes the outward normal derivative of the unknown function on the boundary.

= Far-field boundary condition
{parent=Boundary condition}

A far-field boundary condition prescribes the limiting behavior of a solution as one or more spatial coordinates tend to infinity.

= Sign-decay differential equation
{parent=Ordinary differential equation}

For $y_0>0$, the initial-value problem
$$
y'=-\operatorname{sign}(y),
\qquad y(0)=y_0,
$$
has the continuous piecewise differentiable solution
$$
y(t)=\max\{y_0-t,0\}.
$$
The vector field is not Lipschitz at zero, but the direction on each side prevents any solution from leaving zero, so this solution is unique in the piecewise differentiable class.

= Explicit Euler method for the sign-decay equation
{parent=Sign-decay differential equation}

The iteration $y_{n+1}=y_n-h\operatorname{sign}(y_n)$ agrees with the exact linear descent until the first step past zero. It then either remains at zero or alternates between two values of magnitude at most $h$. Consequently its uniform grid-point error is at most $h$ for arbitrarily long finite time intervals.

= Parameter sensitivity equation
{parent=Ordinary differential equation}

Differentiating a parameter-dependent initial-value problem with respect to its parameter gives a linear inhomogeneous equation for the solution sensitivity, with initial data obtained by differentiating the original initial condition.

= Differential inequality
{parent=Ordinary differential equation}
{wiki}

A differential inequality bounds derivatives rather than specifying them exactly; comparison and integration turn it into bounds on the function.

= Bernoulli differential equation
{parent=Ordinary differential equation}
{c}
{wiki=Bernoulli_differential_equation}

For
$$
y'+P(x)y=Q(x)y^n,\qquad n\ne0,1,
$$
the substitution $z=y^{1-n}$ gives the linear equation
$$
z'+(1-n)Pz=(1-n)Q.
$$

= Tangent addition functional equation
{parent=Ordinary differential equation}

A differentiable real function satisfying
$$
f(x+y)=\frac{f(x)+f(y)}{1-f(x)f(y)}
$$
on an interval has $f(0)=0$ and $f'=C(1+f^2)$, hence $f(x)=\tan(Cx)$ wherever the tangent remains finite.

= Linear ordinary differential equation
{parent=Ordinary differential equation}
{wiki}

A linear ordinary differential equation is linear in the unknown function and its derivatives.

= Homogeneous solution
{parent=Linear ordinary differential equation}
{wiki=Homogeneous_differential_equation}

A homogeneous solution solves the associated linear equation with zero forcing. The difference of any two solutions to the same forced linear equation is homogeneous.

= Particular solution
{parent=Linear ordinary differential equation}
{wiki}

A particular solution is any one solution of a forced linear differential equation. Every solution is the sum of that particular solution and an arbitrary <homogeneous solution>.

= Dominant eigenmode in a forced linear system
{parent=Linear ordinary differential equation}

After diagonalising a constant-coefficient system, each eigenmode satisfies a scalar forced equation. The large-time behaviour is determined by the largest exponential rate whose coefficient does not vanish, including rates introduced by the forcing.

= Second-order linear differential equation
{parent=Linear ordinary differential equation}

A second-order linear equation has the form $y\prime\prime+p(x)y\prime+q(x)y=f(x)$.

= Sturm comparison theorem
{parent=Second-order linear differential equation}
{c}
{wiki=Sturm_comparison_theorem}

If $q_1\leq q_2$, then between consecutive zeros of a nontrivial solution of
$$
y''+q_1(x)y=0
$$
there is a zero of every nontrivial solution of $y''+q_2(x)y=0$, unless the two coefficients agree throughout that interval.

= Wronskian proof of Sturm comparison
{parent=Sturm comparison theorem}

For solutions $\varphi_i''+q_i\varphi_i=0$, the mixed Wronskian
$$
W=\varphi_1'\varphi_2-\varphi_1\varphi_2'
$$
satisfies
$$
W'=(q_2-q_1)\varphi_1\varphi_2.
$$
Its endpoint signs between consecutive zeros of a positive $\varphi_1$ force a zero of $\varphi_2$ unless $q_1=q_2$.

= Power-series solution of a differential equation
{parent=Second-order linear differential equation}
{wiki}

= Kummer differential equation
{parent=Second-order linear differential equation}
{c}
{wiki=Confluent_hypergeometric_function}

Kummer's equation is
$$
xy''+(b-x)y'-ay=0.
$$
Its solution analytic at zero is the confluent hypergeometric function
$$
M(x,a,b)=\sum_{m=0}^{\infty}
\frac{(a)_m}{(b)_m\,m!}x^m.
$$
When $b$ is not an integer, a second local solution is
$$
x^{1-b}M(x,a-b+1,2-b).
$$

= Wronskian
{parent=Second-order linear differential equation}
{c}
{wiki}

= Abel identity
{parent=Wronskian}
{c}
{wiki=Abel%27s_identity}

For $y\prime\prime+py\prime+qy=0$, the Wronskian satisfies $W\prime=-pW$.

= Variation of parameters
{parent=Second-order linear differential equation}
{wiki}

Variation of parameters replaces the constants in a complementary solution by functions to construct a particular solution.

= Reduction of order
{parent=Second-order linear differential equation}
{wiki}

Given one nonzero solution of a homogeneous second-order linear equation, reduction of order constructs a second by writing it as a variable multiple of the first.

= Cauchy-Euler differential equation
{parent=Second-order linear differential equation}
{c}
{wiki=Cauchy–Euler_equation}

A Cauchy–Euler equation has powers of the independent variable matched to derivative order and is solved using power laws or a logarithmic change of variable.

= Euler-Cauchy equation
{parent=Cauchy-Euler differential equation}
{c}
{wiki}

= General solution of an Euler-Cauchy equation
{parent=Cauchy-Euler differential equation}

For
$$
z^2w''+azw'+bw=0,
$$
the power ansatz $w=z^\lambda$ gives the indicial equation $\lambda(\lambda-1)+a\lambda+b=0$. Distinct roots give $C_1z^{\lambda_1}+C_2z^{\lambda_2}$; a repeated root $\lambda$ gives $z^\lambda(C_1+C_2\log z)$ on a chosen logarithm branch.

= Legendre differential equation
{parent=Second-order linear differential equation}
{c}
{wiki=Legendre_differential_equation}

The equation $(1-x^2)y\prime\prime-2xy\prime+\ell(\ell+1)y=0$ has polynomial solutions for nonnegative integer $\ell$.

= Legendre's differential equation
{c}
{synonym}

= Legendre polynomial
{parent=Legendre differential equation}
{c}
{wiki}

The polynomial solution of degree l normalized by P_l(1)=1 is the Legendre polynomial P_l.

= Schläfli contour integral for Legendre polynomials
{parent=Legendre polynomial}
{c}

For a contour enclosing $t$,
$$
P_n(t)=\frac1{2^{n+1}\pi i}
\oint\frac{(z^2-1)^n}{(z-t)^{n+1}}\,dz.
$$

= Debye asymptotic for Legendre polynomials
{parent=Legendre polynomial}
{c}

For fixed $0<\theta<\pi$,
$$
P_n(\cos\theta)
\sim
\sqrt{\frac{2}{\pi n\sin\theta}}
\cos\left(\left(n+\frac12\right)\theta-\frac\pi4\right).
$$

= Change of independent variable in a second-order ODE
{parent=Second-order linear differential equation}

For $z=z(x)$, the second derivative transforms as $w_{xx}=z_x^2w_{zz}+z_{xx}w_z$.

= Integrating factor
{parent=Linear ordinary differential equation}
{wiki}

An integrating factor turns a first-order linear equation into an exact derivative.

= Bounded solution selected by a terminal condition
{parent=Integrating factor}

For a first-order linear equation whose homogeneous solution grows at infinity, boundedness fixes the integration constant by rewriting the particular integral as a tail integral from the current point to infinity.

= Resonance in a differential equation
{parent=Linear ordinary differential equation}

Resonance occurs when forcing overlaps a homogeneous mode, requiring multiplication of the usual particular ansatz by an extra power or logarithm.

= Resonant forcing
{parent=Resonance in a differential equation}
{wiki}

When forcing matches a natural frequency, the oscillator response acquires a linearly growing amplitude.

= Linear system of differential equations
{parent=Linear ordinary differential equation}
{wiki=Linear_differential_equation\#System_of_linear_differential_equations}

A linear differential system has vector form $x\prime=A(t)x+b(t)$.

= Eigenvector method for a differential equation
{parent=Linear system of differential equations}

For a constant coefficient system, eigenvectors split the homogeneous equation into scalar exponential or power-law modes.

= State transition matrix
{parent=Linear system of differential equations}
{wiki}

A state transition matrix maps the state of a homogeneous linear system between two times.

= Complex form of a planar linear system
{parent=Linear system of differential equations}
{wiki}

Encoding two planar components as one complex variable turns a rotation-dilation system into a scalar complex equation.

= Jump condition for an impulse
{parent=Linear ordinary differential equation}

Integrating an equation through a Dirac impulse determines the jump in the derivative or state while nonsingular lower-order terms contribute no jump.

= First integral
{parent=Ordinary differential equation}
{wiki}

A first integral is a function of time, state, and derivatives that remains constant along every solution.

= Real analytic function
{parent=Analysis}
{wiki}

= Residue
{title2=$\operatorname{Res}$}
{parent=Analysis}
{wiki}

The residue of a <meromorphic function> $f$ at an <isolated singularity> $a$ is the coefficient of $(z-a)^{-1}$ in its <Laurent series>. At a simple <pole>,
$$
\operatorname{Res}(f,a)=\lim_{z\to a}(z-a)f(z).
$$

= Residue at a pole of order n
{parent=Residue}

If
$$
f(z)=\frac{g(z)}{(z-a)^n}
$$
with $g$ holomorphic at $a$, then
$$
\operatorname{Res}(f,a)
=\frac{g^{(n-1)}(a)}{(n-1)!}.
$$

= Residue theorem
{parent=Analysis}
{wiki}

If $f$ is <holomorphic function>[holomorphic] on and inside a positively oriented closed contour except for finitely many isolated singularities $a_j$ inside it, then
$$
\oint f(z)\,dz=2\pi i\sum_j\operatorname{Res}(f,a_j).
$$

= Upper-half-plane indentation rule
{parent=Residue theorem}

Suppose a contour follows the <real line>[real axis], closes in the upper half-plane, and avoids each simple real pole $a_j$ by a small semicircle above it. Each indentation is clockwise and tends to
$$
-i\pi\operatorname{Res}(f,a_j).
$$
If the large arc vanishes and there are no enclosed poles away from the real axis, the <Cauchy principal value> is therefore
$$
\operatorname{PV}\int_{-\infty}^{\infty}f(x)\,dx
=i\pi\sum_j\operatorname{Res}(f,a_j).
$$

= Residue cancellation by symmetry
{parent=Residue theorem}
{wiki}

Residues at symmetry-related poles can occur with opposite signs and cancel in the contour sum.

= Symmetric residue summation
{parent=Residue theorem}

Pairing residues at opposite poles can produce an alternating series that converges in a larger half-plane than the corresponding absolutely summed residue series.

= Schwarz lemma
{parent=Analysis}
{c}
{wiki}

= Spectral radius
{title2=$\rho(A)$}
{parent=Analysis}
{wiki}

The spectral radius is the largest modulus of an operator's or matrix's eigenvalues:
$$
\rho(A)=\max_{\lambda\in\sigma(A)}|\lambda|.
$$

= Spherical average
{parent=Analysis}
{wiki}

= Sturm-Liouville theory
{parent=Analysis}
{c}
{wiki}

= Sturm-Liouville problem
{parent=Sturm-Liouville theory}
{c}
{wiki=Sturm%E2%80%93Liouville_theory}

A regular Sturm-Liouville eigenvalue problem has the form
$$
+(p y')'+q y=-\lambda w y
$$
on a bounded interval with self-adjoint boundary conditions and positive weight $w$.

= Dirichlet eigenvalue
{parent=Sturm-Liouville problem}
{c}

A Dirichlet eigenvalue is an eigenvalue for an eigenfunction constrained to vanish at the endpoints of its interval or on the boundary of its domain.

= Sturm-Liouville eigenfunction expansion
{parent=Sturm-Liouville problem}
{c}

For a regular self-adjoint Sturm-Liouville problem, eigenfunctions are orthogonal in the weighted inner product
$$
\langle f,g\rangle_w=\int fgw,
$$
and the coefficient of $y_n$ is $\langle f,y_n\rangle_w/\langle y_n,y_n\rangle_w$.

= Self-adjoint differential operator
{parent=Sturm-Liouville theory}
{wiki}

A differential operator is self-adjoint when integration by parts makes its inner products symmetric on its domain.

= Self-adjoint differential equation
{parent=Self-adjoint differential operator}

A second-order scalar differential equation is in self-adjoint form when it can be written
$$
(p(x)y')'+q(x)y=0.
$$
Multiplication by $y$ and integration by parts then gives an energy identity.

= Solvability condition at a Sturm-Liouville eigenvalue
{parent=Self-adjoint differential operator}

If $L(y_0;\lambda_0)=0$ and $y_0,y$ satisfy homogeneous separated boundary conditions, the Lagrange identity gives
$$
\int y_0L(y;\lambda_0)\,dx
=\int yL(y_0;\lambda_0)\,dx=0.
$$
Thus $L(y;\lambda_0)=f$ can be solved only if $f$ is orthogonal to the null mode $y_0$.

= Leading nonlinear eigenvalue shift in a Sturm-Liouville problem
{parent=Solvability condition at a Sturm-Liouville eigenvalue}

For $L(y;\lambda)=y^{m+1}$ near a normalized simple null mode $y_0$, put $y=\varepsilon y_0+O(\varepsilon^2)$ and $\lambda-\lambda_0=\varepsilon^m\mu$. Orthogonality to $y_0$ yields
$$
\mu=\int y_0^{m+2}\,dx+O(\varepsilon).
$$

= Travelling wave
{parent=Analysis}
{wiki}

= Travelling-wave reduction of a reaction-diffusion system
{parent=Travelling wave}

The substitution $U(x,t)=U(x-ct)$ turns time derivatives into $-cU'$ and spatial diffusion into $U''$, reducing a reaction-diffusion PDE to an ODE system.

= Linear spreading speed
{parent=Travelling wave}

The linear spreading speed is obtained from exponentially decaying modes at an unstable state's leading edge and the condition that their spatial exponents become repeated.

= Fisher-KPP minimum wave speed
{parent=Linear spreading speed}
{c}
{wiki=Fisher%27s_equation}

For diffusion coefficient $D$ and leading-edge growth rate $r>0$, a Fisher-KPP-type pulled front has minimum speed $2\sqrt{Dr}$.

= Pulled travelling front
{parent=Linear spreading speed}

A pulled front is selected by growth and diffusion in its small-amplitude leading edge rather than by nonlinear dynamics behind the front.

= Uniqueness of Poisson equation
{parent=Analysis}
{wiki}

= Uniqueness theorem for ordinary differential equations
{parent=Analysis}
{wiki}

= Real analysis
{parent=Analysis}
{wiki}

Real analysis studies limits, continuity, differentiation, integration, and convergence for real-valued functions.

= Lipschitz continuity
{parent=Real analysis}
{wiki}

A function $f$ is Lipschitz continuous when there is a constant $L\geq0$ such that
$$
|f(x)-f(y)|\leq L|x-y|
$$
for all points in its domain.

= Lipschitz continuous
{synonym}

= Lipschitz bound
{parent=Lipschitz continuity}

A Lipschitz bound has the form $|f(x)-f(y)|\leq L|x-y|$. It controls the change of a function by the change of its argument.

= Big O notation
{title2=$O(\mathord\cdot)$}
{parent=Real analysis}
{c}
{wiki=Big_O_notation}

The relation $f(x)=O(g(x))$ near a limit point means that $|f(x)|\leq C|g(x)|$ there for some constant $C$.

= Real line
{parent=Real analysis}
{wiki}

The real line is the set $\mathbb R$ of real numbers equipped with its usual order, metric, and topology.

= Real interval
{parent=Real line}
{wiki=Interval_(mathematics)}

A real interval contains every real number lying between any two of its elements.

= Real intervals
{synonym}

= Closed real interval
{parent=Real interval}
{wiki=Interval_(mathematics)}

A bounded closed real interval has the form $[a,b]=\{x\in\mathbb R:a\leq x\leq b\}$.

= Closed real intervals
{synonym}

= Closed interval
{synonym}

= Closed intervals
{synonym}

= Extreme value theorem
{parent=Real analysis}
{wiki}

A real-valued <continuous function> on a nonempty <compact space> attains both its minimum and its maximum.

= Coercive function
{parent=Real analysis}
{wiki=Coercive_function}

A real-valued function on a normed vector space is coercive when its value tends to positive infinity as the norm of its argument tends to infinity. A continuous coercive function on a finite-dimensional real vector space attains a global minimum.

= Lp norm
{title2=$\lVert\mathord\cdot\rVert_p$}
{parent=Real analysis}
{wiki}

For a measurable function $f$, its $L^p$ norm is
$$
\lVert f\rVert_p=\left(\int |f|^p\right)^{1/p}
$$
when $1\leq p<\infty$, while $\lVert f\rVert_\infty$ is its essential supremum.

= Sequence and series
{parent=Real analysis}
{wiki}

A sequence is an ordered family indexed by the natural numbers; a series studies the partial sums of a sequence of terms.

= Sequence
{parent=Sequence and series}
{wiki}

A sequence is a function whose domain is usually the natural numbers.

= Bounded sequence
{parent=Sequence}
{wiki}

A sequence $(x_n)$ in a metric space is bounded when all its terms lie in some ball of finite radius.

= Binary sequence
{parent=Sequence}
{wiki=Binary_sequence}

A binary sequence is a finite or infinite sequence whose terms belong to $\{0,1\}$.

= Monotone sequence
{parent=Sequence}
{wiki=Monotonic_function#In_sequence_analysis}

A real sequence is increasing when $a_{n+1}\geq a_n$ and decreasing when $a_{n+1}\leq a_n$.

= Limit superior
{title2=$\limsup$}
{parent=Sequence}
{wiki}

The limit superior of a real sequence $(a_n)$ is
$$
\limsup_{n\to\infty}a_n=\lim_{n\to\infty}\sup_{k\geq n}a_k.
$$
It is the largest subsequential limit when the sequence is bounded.

= Series
{parent=Sequence and series}
{wiki=Series_(mathematics)}

A series is the formal or limiting sum of the terms of a <sequence>.

= Partial sum
{title2=$S_n$}
{parent=Series}
{wiki}

For terms $(a_n)$, the $n$th partial sum is $S_n=\sum_{j=1}^n a_j$. A <series> converges when its sequence of partial sums converges.

= Fekete lemma
{parent=Sequence and series}
{c}
{wiki=Fekete%27s_lemma}

If a real sequence is subadditive, $a_{m+n}\leq a_m+a_n$, and is bounded below in the required extended-real sense, then $a_n/n$ converges to $\inf_{k\geq1}a_k/k$.

= Bolzano-Weierstrass theorem
{parent=Sequence and series}
{c}
{wiki}

Every bounded real sequence has a convergent subsequence. One proof repeatedly bisects a closed interval containing infinitely many terms, chooses a nested half containing infinitely many terms, and then chooses indices increasingly from those halves. Their interval diameters tend to zero, so completeness gives convergence.

= Nested interval theorem
{parent=Bolzano-Weierstrass theorem}
{wiki=Nested_intervals}

If $I_1\supseteq I_2\supseteq\cdots$ are nonempty closed bounded intervals, then their intersection is nonempty. If their lengths tend to zero, the intersection consists of exactly one point.

= Unique subsequential limit of a bounded sequence
{parent=Bolzano-Weierstrass theorem}

If a bounded real sequence has the property that every convergent subsequence converges to the same number $L$, then the full sequence converges to $L$. Otherwise, a subsequence stays at least some fixed distance from $L$; Bolzano--Weierstrass gives it a convergent subsubsequence, contradicting the assumed uniqueness.

= Convergent sequence
{parent=Sequence and series}
{wiki}

= Cesaro mean
{title2=$\frac1n\sum_{k=1}^n a_k$}
{parent=Convergent sequence}
{c}
{wiki=Ces%C3%A0ro_summation}

The Cesaro mean of the first $n$ terms of a sequence is their arithmetic average. If $a_n\to L$, then its Cesaro means also tend to $L$.

= Geometric mean of a sequence
{title2=$\sqrt[n]{x_1\cdots x_n}$}
{parent=Convergent sequence}

For a positive sequence converging to $L\geq0$, its cumulative geometric means converge to the same limit.

= Monotone bounded sequence
{parent=Convergent sequence}

An increasing real sequence converges exactly when it is bounded above. For a bounded sequence, its limit is the supremum of its set of terms.

= Fibonacci number
{parent=Sequence and series}
{c}
{wiki}

The Fibonacci numbers satisfy $F_0=0$, $F_1=1$, and $F_{n+1}=F_n+F_{n-1}$.

= Fibonacci determinant identity
{parent=Fibonacci number}
{c}

For nonnegative $n,m,l$,
$$
F_{n+l}F_{n+m}-F_nF_{n+m+l}
=(-1)^nF_mF_l.
$$
Cassini's identity is the adjacent-index special case.

= Pointwise convergence
{parent=Sequence and series}
{wiki}

= Power series
{parent=Sequence and series}
{wiki}

= Radius of convergence
{parent=Power series}
{wiki}

= Cauchy-Hadamard theorem
{parent=Radius of convergence}
{c}
{wiki=Cauchy%E2%80%93Hadamard_theorem}

For $\sum a_nz^n$,
$$
\frac1R=\limsup_{n\to\infty}|a_n|^{1/n},
$$
with the usual conventions for zero and infinity.

= Radius of convergence after powering coefficients
{parent=Radius of convergence}

If $L=\limsup a_n^{1/n}$ for positive coefficients, then the coefficients $a_n^2$ have root limsup $L^2$. Coefficients $a_n^{a_n}$ instead have root terms
$$
a_n^{a_n/n}=\exp\left(\frac{a_n\log a_n}{n}\right),
$$
so their radius depends on the growth of $a_n\log a_n$ relative to $n$.

= Half-plane of convergence of an exponential power series
{parent=Radius of convergence}

If the ordinary power series $\sum_{n\geq0}b_nw^n$ has radius $R\in(0,\infty)$, then
$$
\sum_{n\geq0}b_ne^{nz}
$$
converges for $\operatorname{Re}z<\log R$ and diverges for $\operatorname{Re}z>\log R$. Behavior on the boundary depends on the coefficients.

= Cauchy product
{parent=Power series}
{c}
{wiki}

The Cauchy product has coefficients $c_n=\sum_{k=0}^na_kb_{n-k}$. Absolute convergence permits regrouping and makes its sum the product of the two sums.

= Termwise differentiation of a power series
{parent=Power series}

Inside its radius of convergence, a power series may be differentiated term by term, and the differentiated series has the same radius.

= Leading Taylor term
{parent=Power series}
{wiki}

The first nonzero Taylor term controls a holomorphic function’s local magnitude and angular sign pattern.

= Uniform convergence
{parent=Sequence and series}
{wiki}

A sequence of functions $f_n:X\to Y$ converges uniformly when one index $N$ makes $f_n(x)$ close to the limit simultaneously for every $x\in X$.

= Uniformly Cauchy sequence
{parent=Uniform convergence}
{wiki=Uniform_convergence#Cauchy_criterion}

A sequence of functions is uniformly Cauchy when, for every $\varepsilon>0$, all sufficiently late pairs satisfy
$$
\sup_x d(f_n(x),f_m(x))<\varepsilon.
$$

= Locally uniform convergence
{parent=Uniform convergence}
{wiki=Uniform_convergence\#Local_uniform_convergence}

A sequence $f_n:X\to Y$ converges locally uniformly to $f$ when every point has a neighbourhood on which the convergence is uniform.

= Local uniform convergence on compact subsets
{parent=Locally uniform convergence}

A locally uniform limit of continuous real-valued functions is continuous. Moreover, local uniform convergence is uniform on every compact subset: choose finitely many neighbourhoods from the local uniformity cover and take the largest of their convergence indices.

= Geometric series
{parent=Sequence and series}
{wiki}

= Finite geometric series
{parent=Geometric series}
{wiki=Geometric_series#Finite_series}

For $r\ne1$, a finite geometric series satisfies
$$
\sum_{j=m}^n r^j=r^m\frac{1-r^{n-m+1}}{1-r}.
$$

= Cauchy sequence
{parent=Sequence and series}
{c}
{wiki}

A sequence is Cauchy when its terms become arbitrarily close to one another beyond some index.

= Completeness of the real numbers
{parent=Cauchy sequence}
{wiki=Completeness_of_the_real_numbers}

Every Cauchy sequence of real numbers converges to a real number.

= Conditional convergence
{parent=Sequence and series}
{wiki}

A series is conditionally convergent when it converges but does not converge absolutely.

= Alternating series test
{parent=Conditional convergence}
{wiki}

If $a_n\geq0$ decreases to zero, then $\sum_{n=1}^{\infty}(-1)^na_n$ converges.

= Alternating harmonic series
{parent=Alternating series test}
{wiki=Alternating_harmonic_series}

The alternating harmonic series $\sum_{n\geq1}(-1)^{n+1}/n$ converges. Its even partial sums increase, its odd partial sums decrease, and the two limits agree because their difference is the next term. In particular, every even partial sum is a lower bound and every odd partial sum is an upper bound for the limit.

= Rearrangement of a series
{parent=Conditional convergence}
{wiki=Riemann_series_theorem}

A rearrangement changes the order of a series without changing its multiset of terms. A conditionally convergent real series can be rearranged to approach any prescribed real limit or to diverge.

= P-series
{parent=Sequence and series}
{wiki=Harmonic_series_(mathematics)#P-series}

The series $\sum_{n=1}^{\infty}n^{-p}$ converges exactly when $p>1$.

= Absolute convergence
{parent=Sequence and series}
{wiki}

A series converges absolutely when the series of absolute values converges; absolute convergence implies convergence.

= Squares of a summable positive sequence
{parent=Absolute convergence}

If $a_n\geq0$ and $\sum a_n<\infty$, then $a_n\to0$, so eventually $a_n^2\leq a_n$ and $\sum a_n^2$ converges.

= Geometric means of two summable positive sequences
{parent=Absolute convergence}

The <Cauchy-Schwarz inequality> gives
$$
\sum_n\sqrt{a_nb_n}
\leq\left(\sum_na_n\right)^{1/2}
\left(\sum_nb_n\right)^{1/2}.
$$

= Weighted square roots of a summable sequence
{parent=Absolute convergence}

If $a_n\geq0$, $\sum a_n<\infty$, and $p>1/2$, then
$$
\sum_n\sqrt{a_n}\,n^{-p}
\leq\left(\sum_na_n\right)^{1/2}
\left(\sum_nn^{-2p}\right)^{1/2}<\infty.
$$
The endpoint can fail: $a_n=1/(n(\log n)^2)$ gives the divergent series $\sum1/(n\log n)$.

= Total variation
{parent=Sequence and series}
{wiki}

The total variation of a sequence is $\sum_n|a_{n+1}-a_n|$; convergence alone does not make it finite.

= Harmonic sum
{parent=Sequence and series}
{wiki=Harmonic_number}

Harmonic sums satisfy $\sum_{j=n+1}^{2n}1/j\to\log2$ by integral comparison.

= Asymptotic equivalence
{parent=Sequence and series}
{wiki=Asymptotic_analysis}

The notation $a_n\sim b_n$ means $a_n/b_n\to1$.

= Asymptotic comparison of exponential functions
{parent=Asymptotic equivalence}
{wiki}

Ratios of exponential functions are controlled by comparing their linear growth exponents.

= Stirling formula
{parent=Asymptotic equivalence}
{c}
{wiki}

Stirling’s formula gives n factorial asymptotic to square root of 2 pi n times (n/e)^n.

= Convergent series
{parent=Sequence and series}
{wiki}

A series converges when its sequence of partial sums converges to a finite limit.

= Limit comparison test
{parent=Convergent series}
{wiki}

For positive terms, if the ratio of two sequences tends to a finite positive number, their series either both converge or both diverge.

= Ratio test
{parent=Convergent series}
{wiki}

For nonzero terms, if
$$
\limsup_{n\to\infty}\left|\frac{a_{n+1}}{a_n}\right|<1,
$$
then $\sum a_n$ converges absolutely. If the ratio has a limit greater than one, the terms do not tend to zero and the series diverges.

= Ratio limit implies root limit
{parent=Ratio test}

For a positive sequence, if $a_{n+1}/a_n\to L\geq0$, then
$$
a_n^{1/n}\to L.
$$
For $L>0$ this follows by taking logarithms and applying Cesàro averaging; the case $L=0$ follows from an eventual geometric upper bound.

= Factorial-over-power series
{parent=Ratio test}

For $a_n=n!/n^n$,
$$
\frac{a_{n+1}}{a_n}=\left(\frac n{n+1}\right)^n\to e^{-1}.
$$
Thus $\sum n!/n^n$ converges and $n/(n!)^{1/n}\to e$.

= Uniform limit
{parent=Sequence and series}
{wiki}

A sequence converges uniformly when one index makes the approximation accurate at every point of the domain.

= Calculus
{parent=Real analysis}
{wiki}

Calculus studies local change through derivatives and accumulated change through integrals.

= Floor function
{title2=$\lfloor x\rfloor$}
{parent=Calculus}
{wiki=Floor_and_ceiling_functions}

The floor $\lfloor x\rfloor$ is the greatest <integer> not exceeding the real number $x$.

= Integral
{title2=$\int$}
{parent=Calculus}
{wiki}

An integral accumulates a function over a domain and may be defined as a limit of finite sums.

= Double integral
{title2=$\iint$}
{parent=Integral}
{wiki=Multiple_integral}

A double integral integrates a function over a two-dimensional region. A <change of variables formula> can replace the region and area element using a <Jacobian determinant>.

= Monotonicity of the Lebesgue integral
{parent=Integral}

If measurable functions satisfy $f\leq g$ almost everywhere, then
$$
\int f\,d\mu\leq\int g\,d\mu
$$
whenever the two sides are defined.

= Antiderivative
{parent=Integral}
{wiki}

An antiderivative of $g$ is a <differentiable function> $p$ satisfying $p'=g$.

= Additive constant
{parent=Antiderivative}
{wiki=Constant_of_integration}

Any two antiderivatives of the same function on a connected interval differ by an additive constant.

= Gaussian integral
{title2=$\int_{-\infty}^{\infty}e^{-x^2}\,dx=\sqrt\pi$}
{parent=Integral}
{c}
{wiki}

The Gaussian integral is
$$
\int_{-\infty}^{\infty}e^{-ax^2}\,dx=\sqrt{\frac\pi a},
\qquad a>0.
$$
Squaring the integral and changing to polar coordinates proves the formula.

= Error function
{title2=$\operatorname{erf}$}
{parent=Gaussian integral}
{wiki}

The error function is
$$
\operatorname{erf}(x)=\frac2{\sqrt\pi}\int_0^xe^{-u^2}\,du.
$$
It is odd and tends to $\pm1$ as $x\to\pm\infty$.

= Taylor polynomial
{parent=Calculus}
{c}
{wiki}

= Exponential function
{title2=$e^x$}
{parent=Calculus}
{wiki}

= Gaussian function
{parent=Exponential function}
{c}
{wiki}

A Gaussian function has the form $A\exp[-a(x-b)^2]$ with $a>0$. Its integral and moments reduce by translation and scaling to the <Gaussian integral>.

= Natural logarithm
{title2=$\ln$}
{parent=Exponential function}
{wiki}

The natural logarithm is the inverse of the real <exponential function> on the positive real numbers.

= Hyperbolic cosine
{title2=$\cosh$}
{parent=Exponential function}
{wiki}

The hyperbolic cosine is $\cosh x=(e^x+e^{-x})/2$.

= Hyperbolic sine
{title2=$\sinh$}
{parent=Exponential function}
{wiki}

The hyperbolic sine is $\sinh x=(e^x-e^{-x})/2$.

= Hyperbolic cotangent
{title2=$\coth$}
{parent=Exponential function}
{wiki}

The hyperbolic cotangent is $\coth x=\cosh x/\sinh x$ where $\sinh x\ne0$.

= Hyperbolic secant
{title2=$\operatorname{sech}$}
{parent=Exponential function}
{wiki}

The hyperbolic secant is
$$
\operatorname{sech}x=\frac1{\cosh x}.
$$

= Fundamental theorem of calculus
{parent=Calculus}
{wiki}

= Mean value theorem
{parent=Calculus}
{wiki}

= Rolle theorem
{parent=Calculus}
{c}
{wiki}

= Taylor theorem
{parent=Calculus}
{c}
{wiki}

= Taylor theorem with Lagrange remainder
{parent=Taylor theorem}
{c}

If $f$ is $n$ times differentiable between $a$ and $x$, then for some $\xi$ strictly between them,
$$
f(x)=\sum_{j=0}^{n-1}\frac{f^{(j)}(a)}{j!}(x-a)^j
+\frac{f^{(n)}(\xi)}{n!}(x-a)^n.
$$

= Taylor series
{parent=Taylor theorem}
{wiki}

The Taylor series of a sufficiently regular <function> about $a$ is
$$
\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n.
$$

= Limit of a function
{title2=$\lim$}
{parent=Calculus}
{wiki=Limit_of_a_function}

The limit records the value approached by a function as its argument approaches a point.

= Limit
{synonym}

= Continuous function
{parent=Limit of a function}
{wiki}

A function is continuous at a point when its limit there equals its value.

= Continuous
{synonym}

= Locally constant function
{parent=Continuous function}
{wiki=Locally_constant_function}

A function is locally constant when every point has a neighbourhood on which the function is constant.

= Locally constant
{synonym}

= Intermediate value theorem
{parent=Continuous function}
{wiki}

If $f:[a,b]\to\mathbb R$ is continuous and $y$ lies between $f(a)$ and $f(b)$, then some $c\in[a,b]$ satisfies $f(c)=y$.

= Zero on a line segment
{parent=Intermediate value theorem}

If a continuous real-valued function on a real or complex vector space has opposite signs at two points, restricting it to the line segment between them and applying the intermediate value theorem gives a zero.

= Telescoping increment lemma
{parent=Intermediate value theorem}

If a continuous $g:[0,na]\to\mathbb R$ satisfies $g(na)-g(0)=na$, then the increments
$$
h(x)=g(x+a)-g(x)-a
$$
satisfy $\sum_{k=0}^{n-1}h(ka)=0$. Continuity forces $h$ to vanish somewhere on $[0,(n-1)a]$.

= Continuous bijection of the real line is monotone
{parent=Intermediate value theorem}

Every continuous injective function from a real interval to the real line is strictly monotone. A continuous bijection $\mathbb R\to\mathbb R$ therefore has a continuous inverse.

= Composition of continuous functions
{parent=Continuous function}

If $f$ is continuous at $x$ and $g$ is continuous at $f(x)$, then $g\circ f$ is continuous at $x$.

= Continuity set of a function
{parent=Continuous function}

The continuity set of a function is the set of points where it is continuous. For real functions it is always a $G_\delta$ set.

= Uniformly continuous function
{parent=Continuous function}
{wiki}

A function is uniformly continuous when one input tolerance works at every point of its domain.

= Squeeze theorem
{parent=Limit of a function}
{wiki}

If two functions with the same limit bound a third nearby, the bounded function has that limit too.

= Derivative
{title2=$\frac{df}{dx}$}
{parent=Calculus}
{wiki}

The derivative is the limit of the difference quotient and gives the best linear approximation to local change.

= Differentiation
{synonym}

= Darboux theorem
{parent=Derivative}
{c}
{wiki=Darboux%27s_theorem_(analysis)}

Every derivative has the intermediate-value property, even when the derivative is discontinuous.

= Product rule
{parent=Derivative}
{wiki}

If $f$ and $g$ are differentiable at $a$, then
$$
(fg)'(a)=f'(a)g(a)+f(a)g'(a).
$$

= Leibniz rule
{c}
{parent=Product rule}
{wiki}

The higher-order product rule is
$$
(fg)^{(n)}=\sum_{k=0}^n\binom nk f^{(k)}g^{(n-k)}.
$$

= Nondifferentiable factor with a differentiable nondegenerate product
{parent=Product rule}

At zero, $f(x)=x$ and $g(x)=1+|x|$ give a differentiable product
$$
f(x)g(x)=x+x|x|
$$
with derivative one, although $g$ is not differentiable.

= Quotient rule
{parent=Derivative}
{wiki}

If $f$ and $g$ are <differentiable function>[differentiable] and $g\ne0$, then
$$
\left(\frac fg\right)'
=\frac{f'g-fg'}{g^2}.
$$

= Chain rule
{parent=Derivative}
{wiki}

If $f$ is differentiable at $a$ and $g$ is differentiable at $f(a)$, then
$$
(g\circ f)'(a)=g'(f(a))f'(a).
$$

= Nondifferentiable inner function with a differentiable nondegenerate composite
{parent=Chain rule}

At zero, $f(x)=\sqrt[3]x$ is not differentiable and $g(y)=y^3$ is differentiable, while $g\circ f(x)=x$ has derivative one.

= Second derivative chain rule
{parent=Chain rule}

For twice differentiable functions,
$$
(g\circ f)''(a)
=g''(f(a))[f'(a)]^2+g'(f(a))f''(a).
$$

= Monotone function
{parent=Calculus}
{wiki=Monotonic_function}

A function is increasing when $x\le y$ implies $f(x)\le f(y)$.

= Strictly increasing function
{parent=Monotone function}
{wiki=Monotonic_function#Strictly_monotonic_functions}

A function is strictly increasing when $x<y$ implies $f(x)<f(y)$. It is therefore <injective function>[injective] and each value has at most one preimage.

= Lebesgue theorem on differentiability of monotone functions
{parent=Monotone function}
{c}
{wiki=Differentiability_of_monotone_functions}

Every real-valued <monotone function> on a <real interval> is <differentiable function>[differentiable] with a finite <derivative> at <almost everywhere>[almost every] point.

= Multivariable calculus
{parent=Calculus}
{wiki}

Multivariable calculus extends differentiation and integration to functions of several variables.

= Partial derivative
{title2=$\partial$}
{parent=Multivariable calculus}
{wiki}

A partial derivative differentiates a multivariable <function> with respect to one variable while holding the others fixed.

= Gradient
{title2=$\nabla f$}
{parent=Partial derivative}
{wiki}

The gradient of a scalar-valued <differentiable function> is the <vector> of its first partial derivatives.

= Total differential
{title2=$df$}
{parent=Partial derivative}
{wiki=Total_derivative}

For a differentiable scalar function of several variables, the total differential is
$$
df=\sum_i\frac{\partial f}{\partial x_i}\,dx_i.
$$

= Hessian matrix
{title2=$\operatorname{Hess} f$}
{parent=Multivariable calculus}
{wiki}

The Hessian is the symmetric matrix of second partial derivatives; definiteness classifies <nondegenerate critical points>.

= Nondegenerate critical point
{parent=Hessian matrix}
{wiki=Morse_theory}

A critical point of a twice differentiable function is nondegenerate when its <Hessian matrix> is invertible there. A positive-definite Hessian gives a strict local minimum, a negative-definite Hessian gives a strict local maximum, and an indefinite Hessian gives a saddle point.

= Vector calculus
{parent=Multivariable calculus}
{wiki}

Vector calculus studies differentiation and integration of scalar and vector fields.

= Vector field
{parent=Vector calculus}
{wiki}

A vector field assigns a vector to every point of its domain.

= Helicity conservation by tangent boundary conditions
{parent=Vector calculus}

If a helicity density evolves as a sum of directional derivatives along divergence-free fields tangent to the boundary, its volume integral is conserved because both terms become vanishing boundary fluxes.

= Divergence
{title2=$\nabla\cdot$}
{parent=Vector calculus}
{wiki}

For $F=(F_1,F_2,F_3)$ in Cartesian coordinates,
$$
\nabla\cdot F=\partial_iF_i
=\frac{\partial F_1}{\partial x}
+\frac{\partial F_2}{\partial y}
+\frac{\partial F_3}{\partial z}.
$$

= Curl
{title2=$\nabla\times$}
{parent=Vector calculus}
{wiki}

The curl of $F$ has components
$$
(\nabla\times F)_i=\epsilon_{ijk}\partial_jF_k.
$$

= Curl of the curl identity
{parent=Curl}

For a twice differentiable vector field $A$,
$$
\nabla\times(\nabla\times A)
=\nabla(\nabla\cdot A)-\nabla^2A.
$$

= Laplacian
{title2=$\Delta$}
{parent=Vector calculus}
{wiki=Laplace_operator}

The Laplacian of a twice differentiable scalar field is the divergence of its gradient, $\nabla^2f=\nabla\cdot\nabla f$; it acts componentwise on vector fields.

= Divergence and curl of a cross product
{parent=Vector calculus}

For smooth vector fields $F,G$,
$$
\nabla\cdot(F\times G)
=G\cdot(\nabla\times F)-F\cdot(\nabla\times G)
$$
and
$$
\nabla\times(F\times G)
=F(\nabla\cdot G)-G(\nabla\cdot F)
+(G\cdot\nabla)F-(F\cdot\nabla)G.
$$
Both follow by contracting two Levi-Civita symbols and applying the product rule.

= Levi-Civita symbol
{parent=Vector calculus}
{c}
{wiki}

In three dimensions, $\epsilon_{ijk}$ is zero when indices repeat and is the sign of the permutation $(i,j,k)$ otherwise.

= Contraction of two Levi-Civita symbols
{parent=Levi-Civita symbol}

Contracting one index gives
$$
\epsilon_{ijk}\epsilon_{ipq}
=\delta_{jp}\delta_{kq}-\delta_{jq}\delta_{kp}.
$$

= Lagrange identity for the cross product
{parent=Levi-Civita symbol}
{c}
{wiki=Lagrange%27s_identity}

For three-dimensional vectors,
$$
|x\times y|^2=|x|^2|y|^2-(x\cdot y)^2.
$$
The corresponding identity in $\mathbb R^n$ is
$$
|x|^2|y|^2-(x\cdot y)^2
=\frac12\sum_{i,j}(x_iy_j-x_jy_i)^2.
$$

= Vector triple product
{parent=Levi-Civita symbol}
{wiki=Triple_product}

Contracting two Levi-Civita symbols gives
$$
a\times(b\times c)=b(a\cdot c)-c(a\cdot b).
$$

= Feasible inner products with two unit vectors
{parent=Vector calculus}

For linearly independent unit vectors $y,z$ with $c=y\cdot z$, a pair
$$
p=x\cdot y,\qquad q=x\cdot z
$$
is attained by some unit vector $x$ exactly when
$$
\frac{p^2-2cpq+q^2}{1-c^2}\leq1.
$$
The left-hand side is the squared norm of the least-norm vector having those two inner products.

= Conservative vector field
{parent=Vector calculus}
{wiki}

= Poincare lemma
{c}
{parent=Conservative vector field}
{wiki=Poincar%C3%A9_lemma}

On a simply connected open subset of $\mathbb R^3$, every continuously differentiable curl-free vector field is the gradient of a scalar potential.

= Divergence theorem
{parent=Vector calculus}
{wiki}

= Fundamental theorem for line integrals
{parent=Vector calculus}
{wiki}

= Stokes theorem
{parent=Vector calculus}
{c}
{wiki}

= Surface integral
{parent=Vector calculus}
{wiki}

A surface integral uses the area element induced by a parametrisation to integrate over a surface.

= Parametrized surface
{parent=Surface integral}
{wiki}

A parametrised surface $r(u,v)$ has area element $|r_u\times r_v|\,du\,dv$.

= Surface area of a graph
{parent=Surface integral}
{wiki}

The graph $z=f(x,y)$ has area element $\sqrt{1+f_x^2+f_y^2}\,dx\,dy$.

= Flux integral
{parent=Surface integral}
{wiki}

A flux integral $\int_SF\cdot n\,dS$ measures flow through an oriented surface.

= Green theorem
{parent=Vector calculus}
{c}
{wiki}

Green’s theorem converts circulation around a positively oriented planar boundary into the area integral of scalar curl.

= Line integral
{parent=Vector calculus}
{wiki}

A line integral integrates a scalar or tangential vector-field component along a curve.

= Tensor divergence theorem
{parent=Vector calculus}
{wiki}

Applying the divergence theorem componentwise to $T_{ij}v_i$ yields a boundary traction term and a volume contraction.

= Integration by parts for tensor fields
{parent=Tensor divergence theorem}
{wiki}

Tensor integration by parts transfers a derivative between tensor factors and introduces the boundary contraction with the outward normal.

= Jacobian matrix
{title2=$J_f$}
{parent=Multivariable calculus}
{c}
{wiki}

The Jacobian matrix contains all first partial derivatives of a coordinate transformation.

= Jacobian determinant
{title2=$\det J_f$}
{c}
{parent=Jacobian matrix}
{wiki=Jacobian_matrix_and_determinant}

The Jacobian determinant is the <determinant> of the square <Jacobian matrix>. Its absolute value is the local volume-scaling factor in the <change of variables formula>.

= Change of variables formula
{parent=Multivariable calculus}
{wiki}

The change-of-variables formula multiplies an integral by the absolute determinant of the Jacobian.

= Polar coordinates
{parent=Change of variables formula}
{wiki}

Polar coordinates use $x=r\cos\theta$, $y=r\sin\theta$ and area element $r\,dr\,d\theta$.

= Cardioid
{parent=Polar coordinates}
{wiki}

A cardioid is an epicycloid with one cusp; a standard polar equation is $r=a(1+\cos\theta)$.

= Hyperspherical coordinates
{parent=Change of variables formula}
{wiki}

Hyperspherical coordinates extend polar coordinates to $n$ dimensions with one radius and $n-1$ angles.

= Spherical coordinate system
{parent=Hyperspherical coordinates}
{wiki=Spherical_coordinate_system}

Spherical coordinates $(r,\theta,\phi)$ in $\mathbb R^3$ use
$$
(x,y,z)=(r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta).
$$

= Scale factors of orthogonal coordinates
{parent=Spherical coordinate system}
{wiki=Orthogonal_coordinates}

If orthogonal coordinates $q_i$ satisfy
$$
d\mathbf x=\sum_i h_i\mathbf e_i\,dq_i,
$$
then $h_i=|\partial\mathbf x/\partial q_i|$ are their scale factors.

= Curl in spherical coordinates
{title2=$\nabla\times$}
{parent=Spherical coordinate system}
{wiki=Del_in_cylindrical_and_spherical_coordinates}

For $\mathbf A=A_r\mathbf e_r+A_\theta\mathbf e_\theta+A_\phi\mathbf e_\phi$, the spherical-coordinate curl follows from the orthogonal-coordinate scale factors $1,r,r\sin\theta$.

= Volume of an n-ball
{parent=Hyperspherical coordinates}
{wiki}

The radius-$R$ ball in $\mathbb R^n$ has volume $\pi^{n/2}R^n/\Gamma(n/2+1)$.

= Surface area of an n-sphere
{parent=Hyperspherical coordinates}
{wiki}

The boundary of the radius-$R$ ball has area $2\pi^{n/2}R^{n-1}/\Gamma(n/2)$.

= Differentiable map
{parent=Multivariable calculus}
{wiki=Differentiable_function}

A map is differentiable at $x$ when it differs from an affine map with linear part $Df_x$ by $o(\lVert h\rVert)$ at $x+h$.

= Frechet derivative
{parent=Differentiable map}
{c}
{wiki=Fr%C3%A9chet_derivative}

The Frechet derivative of $f$ at $a$ is the unique <linear map> $Df_a$ such that
$$
\frac{\lVert f(a+h)-f(a)-Df_a(h)\rVert}{\lVert h\rVert}\to0.
$$

= Continuously differentiable function
{parent=Differentiable map}
{wiki=Differentiable_function\#Continuously_differentiable_functions}

A function is continuously differentiable when its derivative exists and varies continuously.

= Continuously differentiable
{synonym}

= Invertible linear map
{parent=Differentiable map}
{wiki=Invertible_linear_map}

An invertible linear map is a linear bijection; its inverse is also linear.

= Mean value inequality
{parent=Differentiable map}
{wiki=Mean_value_theorem\#Mean_value_theorem_for_vector-valued_functions}

If the line segment from $x$ to $y$ lies in the domain of a differentiable map and $\lVert Df_z\rVert\leq M$ along it, then
$$
\lVert f(y)-f(x)\rVert\leq M\lVert y-x\rVert.
$$

= Zero derivative on a connected open set
{parent=Mean value inequality}

A differentiable map on a connected open subset of Euclidean space whose derivative vanishes everywhere is constant. The <mean value inequality> makes it constant on every ball in the domain, hence locally constant; connectedness then makes the value global.

= Inverse function theorem
{parent=Differentiable map}
{wiki}

If a continuously differentiable map has invertible derivative at a point, it is a local continuously differentiable diffeomorphism there.

= Local diffeomorphism
{parent=Inverse function theorem}
{wiki}

A local diffeomorphism restricts near every point to a diffeomorphism onto an open subset. In particular, every local diffeomorphism is an open map.

= Open map
{parent=Local diffeomorphism}
{wiki=Open_and_closed_maps}

An open map sends every open set to an open set.

= Implicit function theorem
{parent=Inverse function theorem}
{wiki}

If $F(x_0,y_0)=0$ and the partial derivative with respect to $y$ is invertible at $(x_0,y_0)$, then the nearby zero set is the graph $y=g(x)$ of a continuously differentiable function.

= Implicit differentiation
{parent=Implicit function theorem}
{wiki}

Implicit differentiation differentiates an identity $F(x,y(x))=0$ and uses the <chain rule> to solve for derivatives of the implicitly defined function. In one dimension, $y'=-F_x/F_y$ when $F_y\ne0$.

= Symmetry in integration
{parent=Calculus}
{wiki}

If an integrand is odd under a measure-preserving symmetry of the domain, its integral is zero.

= Integration by parts
{parent=Calculus}
{wiki}

Integration by parts is the integrated product rule: $\int u\,dv=uv-\int v\,du$.

= Hyperbolic tangent
{parent=Calculus}
{wiki}

The hyperbolic tangent is sinh divided by cosh and approaches plus or minus one at the two infinities.

= Small-argument expansion of the hyperbolic tangent
{title2=$\tanh x=x+O(x^3)$}
{parent=Hyperbolic tangent}

The <taylor series> at zero begins
$$
\tanh x=x-\frac{x^3}{3}+O(x^5),
$$
so $\tanh x\sim x$ as $x\to0$.

= Hyperbolic Pythagorean identity
{parent=Hyperbolic tangent}
{wiki=Hyperbolic_functions#Useful_relations}

The identity $\cosh^2x-\sinh^2x=1$ implies
$$
1-\tanh^2x=\operatorname{sech}^2x,
\qquad
\coth^2x-\operatorname{csch}^2x=1.
$$

= Even function
{parent=Calculus}
{wiki}

An even function satisfies $f(-x)=f(x)$; its <Laurent series> or <taylor series> contains only even powers.

= Even
{synonym}

= Odd function
{parent=Calculus}
{wiki}

An odd function satisfies $f(-x)=-f(x)$; its <Laurent series> or <taylor series> contains only odd powers.

= Odd
{synonym}

= Riemann integration
{parent=Real analysis}
{wiki=Riemann_integral}

Riemann integration approximates area by upper and lower sums over finite partitions.

= Continuous approximation of a Riemann-integrable function
{parent=Riemann integration}

Every Riemann-integrable function on a compact interval can be approximated in $L^1$ by continuous functions. Consequently their integrals converge uniformly over all subintervals.

= Riemann integral
{parent=Riemann integration}
{c}
{wiki}

= Monotonicity of the Riemann integral
{parent=Riemann integral}

If Riemann-integrable functions satisfy $f(x)\leq g(x)$ throughout $[a,b]$, then
$$
\int_a^bf(x)\,dx\leq\int_a^bg(x)\,dx.
$$
This follows directly by comparing their lower and upper Darboux sums, or their Riemann sums on a common sequence of refining partitions.

= Riemann integrability criterion
{parent=Riemann integration}
{c}
{wiki=Riemann_integral}

A bounded function is Riemann integrable exactly when for every $\varepsilon>0$ some partition $P$ satisfies $U(f,P)-L(f,P)<\varepsilon$.

= Continuous functions are Riemann integrable
{parent=Riemann integrability criterion}
{c}

A continuous function on a compact interval is uniformly continuous. A partition with sufficiently small mesh then makes the oscillation on every subinterval small, so its upper and lower Darboux sums can be made arbitrarily close.

= Darboux sum
{parent=Riemann integration}
{wiki}

A lower Darboux sum uses the infimum of a function on each partition interval, while an upper sum uses the supremum.

= Upper and lower Darboux integrals
{parent=Darboux sum}

For a bounded function,
$$
\overline{\int_a^b}f=\inf_PU(f,P),
\qquad
\underline{\int_a^b}f=\sup_PL(f,P).
$$
The function is Riemann integrable exactly when these values agree.

= Improper integration by truncation
{parent=Riemann integration}

For a nonnegative unbounded function, one truncation convention sets $f_r=\min(f,r)$ and asks that every $f_r$ be Riemann integrable and that $\int f_r$ have a finite limit as $r\to\infty$.

= First occurrence of a decimal digit
{parent=Improper integration by truncation}

Under uniform length on $[0,1]$, the position $K$ of the first occurrence of a fixed decimal digit has
$$
\Pr(K=k)=\frac1{10}\left(\frac9{10}\right)^{k-1}.
$$
The set of expansions in which the digit never occurs has length zero, and
$$
\mathbb E K=\sum_{k\geq1}\frac{k}{10}\left(\frac9{10}\right)^{k-1}=10.
$$

= Supremum
{title2=$\sup$}
{parent=Real analysis}
{wiki=Infimum_and_supremum}

The supremum of a set of real numbers is its least upper bound.

= Suprema
{synonym}

= Convex function
{parent=Real analysis}
{wiki}

A function is convex when its value on a line segment is at most the corresponding affine interpolation of endpoint values.

= Pointwise maximum of convex functions
{parent=Convex function}

The pointwise maximum of finitely many convex functions is convex because
$$
\max_i f_i(tx+(1-t)y)
\leq t\max_i f_i(x)+(1-t)\max_i f_i(y).
$$

= Softplus function
{title2=$\log(1+e^x)$}
{parent=Convex function}
{wiki=Softplus}

The softplus function $s(x)=\log(1+e^x)$ is convex because
$$
s''(x)=\frac{e^x}{(1+e^x)^2}>0.
$$

= Absolute value function
{title2=$|x|$}
{parent=Convex function}
{wiki=Absolute_value}

The absolute value function is convex by the triangle inequality:
$$
|tx+(1-t)y|\leq t|x|+(1-t)|y|.
$$

= Convex
{synonym}

= Strongly convex function
{parent=Convex function}
{wiki=Convex_function#Strongly_convex_functions}

A differentiable function is $\alpha$-strongly convex when
$$
f(y)\geq f(x)+\nabla f(x)\cdot(y-x)
+\frac\alpha2\|y-x\|^2.
$$

= Strictly convex function
{parent=Convex function}
{wiki=Convex_function#Strictly_convex_functions}

A function $f$ on a <convex set> is strictly convex when
$$
f(tx+(1-t)y)<tf(x)+(1-t)f(y)
$$
whenever $x\ne y$ and $0<t<1$.

= Strictly convex
{synonym}

= Uniqueness of a minimizer of a strictly convex function
{parent=Strictly convex function}

A strictly convex function has at most one minimizer: if distinct points attained the same minimum, every strict convex combination of them would have a smaller value.

= Convexity domain of x cubed plus y cubed plus Axy
{parent=Convex function}

The <hessian matrix> of
$$
f(x,y)=x^3+y^3+Axy
$$
is positive semidefinite exactly when
$$
x\geq0,\qquad y\geq0,\qquad 36xy\geq A^2.
$$
This region is convex: for $A\ne0$ it is the epigraph of $A^2/(36x)$ in the positive quadrant, and for $A=0$ it is the closed first quadrant.

= Perspective function
{parent=Convex function}
{wiki}

The perspective of f is g(t,x)=t f(x/t) for positive t and preserves convexity.

= Subgradient inequality
{parent=Convex function}
{wiki}

A vector is a subgradient when its supporting affine function lies below the convex function.

= Jensen inequality
{parent=Convex function}
{c}
{wiki=Jensen%27s_inequality}

For a <convex function> $f$ and an integrable <random variable> $X$,
$$
f(\mathbb E X)\leq\mathbb E[f(X)].
$$
The inequality is reversed when $f$ is <concave>.

= Concave function
{parent=Real analysis}
{wiki=Concave_function}

A function $f$ is concave exactly when $-f$ is <convex>. Equivalently,
$$
f(tx+(1-t)y)\geq tf(x)+(1-t)f(y)
$$
for $0\leq t\leq1$.

= Concave
{synonym}

= Concavity
{synonym}

= Strictly concave function
{parent=Concave function}
{wiki=Concave_function#Strict_concavity}

A function is strictly concave when its concavity inequality is strict for distinct points and coefficients strictly between zero and one. A strictly concave function has at most one maximizer on a convex set.

= Product maximizer on a compact convex subset of the positive orthant
{parent=Strictly concave function}

On the positive orthant,
$$
F(x)=\sum_{j=1}^n\log x_j
$$
is strictly concave. Its unique maximizer $x^*$ on a compact convex feasible set satisfies the first-order inequality
$$
\nabla F(x^*)\cdot(x-x^*)\leq0,
$$
equivalently
$$
\sum_{j=1}^n\frac{x_j}{x_j^*}\leq n.
$$

= Measure theory
{parent=Real analysis}
{wiki}

Measure theory supplies a rigorous language for size and integration on general spaces.

= Measure density
{title2=$\rho_{\mu,A}(x)$}
{parent=Measure theory}
{wiki=Lebesgue%27s_density_theorem}

For a measurable set $A\subseteq\mathbb R^n$, its density at $x$ with respect to a locally finite measure $\mu$ is
$$
\rho_{\mu,A}(x)=\lim_{r\downarrow0}\frac{\mu(A\cap B_r(x))}{\mu(B_r(x))}
$$
when this <limit> exists.

= Lebesgue differentiation theorem
{parent=Measure density}
{c}
{wiki}

If $f$ is locally <Lebesgue integrable function>[Lebesgue integrable] on $\mathbb R^n$, then for <almost everywhere>[Lebesgue almost every] $x$,
$$
\lim_{r\downarrow0}\frac1{|B_r(x)|}\int_{B_r(x)}f(y)\,dy=f(x).
$$

= Lebesgue point
{parent=Lebesgue differentiation theorem}
{c}
{wiki}

A point $x\in\mathbb R^n$ is a Lebesgue point of a locally <Lebesgue integrable function> $f$ when
$$
\lim_{r\downarrow0}\frac1{|B_r(x)|}
\int_{B_r(x)}|f(y)-f(x)|\,dy=0.
$$
The <Lebesgue differentiation theorem> says that almost every point is a Lebesgue point.

= Differentiation of an indefinite Lebesgue integral
{parent=Lebesgue differentiation theorem}

If $g$ is locally <Lebesgue integrable function>[Lebesgue integrable] and
$$
G(x)=\int_a^x g(t)\,dt,
$$
then $G'(x)=g(x)$ at every <Lebesgue point> of $g$, and hence <almost everywhere>.

= Lebesgue density theorem
{parent=Measure density}
{c}
{wiki=Lebesgue%27s_density_theorem}

For every <Lebesgue measurable set> $A\subseteq\mathbb R^n$, its <measure density>[Lebesgue density] is $1$ at almost every point of $A$ and $0$ at almost every point of its complement.

= Lebesgue measurable set
{parent=Measure theory}
{wiki=Lebesgue_measure}

A Lebesgue measurable set is a member of the completion of the Borel sigma-algebra with respect to <Lebesgue measure>. In particular, it differs from a <Borel set> by a null set.

= Almost everywhere
{parent=Measure theory}
{wiki=Almost_everywhere}

A property holds almost everywhere with respect to a <measure> when the set of points where it fails has measure zero.

= Lebesgue almost everywhere
{synonym}

= Lebesgue integrable function
{parent=Measure theory}
{wiki=Lebesgue_integration}

A measurable function $f$ is Lebesgue integrable when $\int |f|\,d\lambda<\infty$.

= Conditional expectation
{title2=$\mathbb E[X\mid\mathcal G]$}
{parent=Measure theory}
{wiki}

For an integrable random variable $X$ and a sub-sigma-algebra $\mathcal G$, the conditional expectation $\mathbb E[X\mid\mathcal G]$ is the almost-everywhere unique $\mathcal G$-measurable integrable random variable satisfying
$$
\int_G\mathbb E[X\mid\mathcal G],d\mu=\int_GX,d\mu
$$
for every $G\in\mathcal G$.

= Tower property of conditional expectation
{title2=$\mathbb E[\mathbb E[X\mid\mathcal G]]=\mathbb E[X]$}
{parent=Conditional expectation}
{wiki=Law_of_total_expectation}

Taking expectation after conditioning recovers the original expectation. More generally, if $\mathcal H\subseteq\mathcal G$, then
$$
\mathbb E[\mathbb E[X\mid\mathcal G]\mid\mathcal H]
=\mathbb E[X\mid\mathcal H].
$$

= Fatou lemma
{c}
{parent=Measure theory}
{wiki=Fatou%27s_lemma}

For nonnegative measurable functions,
$$
\int\liminf_nf_n\,d\mu\leq\liminf_n\int f_n\,d\mu.
$$

= Proof of Fatou lemma
{parent=Fatou lemma}

Set
$$
g_n=\inf_{k\geq n}f_k.
$$
Then $0\leq g_n\uparrow\liminf_k f_k$. The <monotone convergence theorem> and $g_n\leq f_k$ for every $k\geq n$ give
$$
\int\liminf_kf_k\,d\mu
=\lim_n\int g_n\,d\mu
\leq\lim_n\inf_{k\geq n}\int f_k\,d\mu
=\liminf_k\int f_k\,d\mu.
$$

= Strict inequality in Fatou lemma
{parent=Fatou lemma}

On $([0,1],\lambda)$, the functions
$$
f_n=n\mathbf1_{(0,1/n)}
$$
converge pointwise to zero but satisfy $\int f_n\,d\lambda=1$. Hence the two sides of <Fatou lemma> can be zero and one.

= Ergodic theory
{parent=Measure theory}
{wiki}

Ergodic theory studies the long-time statistical behaviour of <measure-preserving transformation>[measure-preserving transformations].

= Measure-preserving transformation
{title2=$T$}
{parent=Ergodic theory}
{wiki}

A measurable transformation $T:X\to X$ preserves a measure $\mu$ when $\mu(T^{-1}A)=\mu(A)$ for every measurable set $A$.

= Invariant sigma-algebra
{title2=$\mathcal I$}
{parent=Measure-preserving transformation}

The invariant sigma-algebra of $T$ consists, modulo null sets, of measurable sets $A$ satisfying $T^{-1}A=A$.

= Ergodic measure-preserving transformation
{parent=Measure-preserving transformation}
{wiki=Ergodicity}

A <measure-preserving transformation> is ergodic when every set in its <invariant sigma-algebra> has measure zero or full measure. Equivalently, every integrable invariant function is almost everywhere constant.

= Ergodic transformation
{synonym}

= Irrational rotation of the circle
{parent=Ergodic measure-preserving transformation}
{wiki=Irrational_rotation}

For irrational $\alpha$, the circle rotation $T(x)=x+\alpha\pmod1$ preserves <Lebesgue measure> and is <ergodic measure-preserving transformation>[ergodic].

= Everywhere interval frequency under an irrational rotation
{parent=Irrational rotation of the circle}

Every orbit of an <irrational rotation of the circle> visits a half-open interval $I$ with limiting frequency equal to its length. One proof sandwiches the orbit of an arbitrary point between inner and outer intervals along a nearby orbit for which the <Birkhoff ergodic theorem> holds.

= Birkhoff ergodic theorem
{c}
{parent=Ergodic theory}
{wiki}

For a <measure-preserving transformation> $T$ and $f\in L^1(\mu)$, the ergodic averages
$$
A_nf=\frac1n\sum_{j=0}^{n-1}f\circ T^j
$$
converge almost everywhere to $\mathbb E[f\mid\mathcal I]$, the <conditional expectation> on the <invariant sigma-algebra>. The limit has the same integral as $f$.

= L1 convergence in the Birkhoff ergodic theorem on a finite measure space
{title2=$A_nf\to f^*$ in $L^1$}
{parent=Birkhoff ergodic theorem}

On a <finite measure> space, the convergence in the <Birkhoff ergodic theorem> also holds in $L^1$. Prove it first for bounded truncations by the <dominated convergence theorem>, then use the $L^1$ contraction $\|A_nh\|_1\leq\|h\|_1$ and <Fatou lemma> to remove the truncation.

= Measure
{parent=Measure theory}
{wiki}

A measure on a <sigma-algebra> is a nonnegative, countably additive set function that assigns zero to the empty set.

= Measure space
{title2=$(X,\mathcal M,\mu)$}
{parent=Measure}
{wiki}

A measure space is a set $X$, a <sigma-algebra> $\mathcal M$ on it, and a <measure> $\mu$ defined on $\mathcal M$.

= Pushforward measure
{title2=$f_*\mu$}
{parent=Measure}
{wiki}

For a measurable map $f:X\to Y$, the pushforward of $\mu$ is the measure
$$
(f_*\mu)(B)=\mu(f^{-1}(B)).
$$
It is the distribution induced on $Y$ by mapping points distributed according to $\mu$ through $f$.

= Change of measure
{parent=Measure}
{wiki}

If a measure $\nu$ has density $w=d\nu/d\mu$ with respect to $\mu$, then
$$
\int g\,d\nu=\int gw\,d\mu.
$$
This identity is the basis of likelihood ratios and <importance sampling>.

= Haar measure
{parent=Measure}
{c}
{wiki}

Every locally compact topological group has a nonzero translation-invariant regular measure, unique up to scale. On a compact group it can be normalized to have total mass one.

= Countable additivity
{parent=Measure}
{wiki=Measure_(mathematics)#Definition}

For pairwise disjoint measurable sets $E_n$, countable additivity means
$$
\mu\!\left(\bigcup_{n=1}^{\infty}E_n\right)
=\sum_{n=1}^{\infty}\mu(E_n).
$$

= Finite measure
{parent=Measure}
{wiki=Finite_measure}

A measure $\mu$ on $X$ is finite when $\mu(X)<\infty$.

= Sigma-finite measure
{parent=Measure}
{wiki=Sigma-finite_measure}

A <measure> is sigma-finite when its space is a countable <set union>[union] of measurable sets of finite measure.

= Indicator function
{title2=$\mathbf 1_E$}
{parent=Measure theory}
{wiki=Indicator_function}

The indicator function of a set $E$ is $\mathbf1_E(x)=1$ for $x\in E$ and $0$ otherwise.

= Indicator vector
{parent=Indicator function}
{wiki=Indicator_vector}

The indicator vector of a subset $E\subseteq\{1,\ldots,n\}$ has coordinate $1$ on $E$ and coordinate $0$ outside $E$.

= Sigma-algebra
{parent=Measure theory}
{wiki}

A sigma-algebra on $\Omega$ contains $\Omega$ and is closed under complements and countable unions.

= Borel set
{parent=Sigma-algebra}
{wiki=Borel_set}

A Borel set in a <topological space> belongs to the smallest <sigma-algebra> containing every <open set>.

= Pi-system
{parent=Sigma-algebra}
{wiki=Pi-system}

A pi-system is a nonempty family of subsets closed under finite intersections.

= Dynkin system
{c}
{parent=Sigma-algebra}
{wiki=Dynkin_system}

A Dynkin system contains $\Omega$, is closed under complements, and is closed under countable disjoint unions. Equivalently, it contains differences $B\setminus A$ whenever $A\subseteq B$ are members.

= Dynkin lemma
{c}
{parent=Dynkin system}
{wiki=Dynkin_system\#Dynkin's_π-λ_theorem}

Every Dynkin system containing a pi-system $\mathcal P$ also contains the sigma-algebra $\sigma(\mathcal P)$. Equivalently, the Dynkin system generated by a pi-system equals the sigma-algebra it generates.

= Fubini's theorem
{parent=Measure theory}
{c}
{wiki}

Fubini's theorem permits the order of integration of an absolutely integrable function on a product measure space to be exchanged.

= Sigma-finite uniqueness theorem for measures
{parent=Measure theory}
{wiki=Pi-system#Significance_in_probability_theory}

If two measures agree on a generating $\pi$-system and the space is a countable union of members having finite common measure, then they agree on the generated $\sigma$-algebra.

= Lebesgue measure
{parent=Sigma-finite uniqueness theorem for measures}
{c}
{wiki}

Lebesgue measure is the unique translation-invariant $\sigma$-finite Borel measure on $\mathbb R$ normalized by $\lambda((0,1])=1$. Translation invariance follows by comparing translated measure with $\lambda$ on half-open intervals and applying uniqueness.

= Completeness of Lebesgue measure
{parent=Lebesgue measure}
{c}

Every subset of a <Lebesgue measure>[Lebesgue-null] set is <Lebesgue measurable set>[Lebesgue measurable] and has measure zero.

= Lebesgue space
{parent=Measure theory}
{c}
{wiki=Lp_space}

The space $L^p$ consists of measurable functions with finite $p$-norm, identified when they agree almost everywhere. This quotient is necessary because the integral seminorm vanishes on functions supported on null sets.

= Riesz-Fischer theorem
{parent=Lebesgue space}
{c}
{wiki}

For $1\leq p\leq\infty$, the Lebesgue space $L^p$ is complete in its usual norm.

= Banach intersection of L1 and L2
{parent=Lebesgue space}
{c}

The intersection $L^1\cap L^2$ is Banach for $\lVert f\rVert_1+\lVert f\rVert_2$, but it is generally incomplete when equipped with the $L^1$ norm alone. Truncations of an $L^1$ function outside $L^2$ provide an $L^1$-Cauchy sequence with no limit in the intersection.

= Approximation by nonnegative simple functions
{parent=Measure theory}
{wiki=Simple_function}

Every nonnegative measurable function is the pointwise increasing limit of nonnegative simple functions, and its integral is the supremum of their integrals.

= Monotone convergence theorem
{parent=Measure theory}
{wiki=Monotone_convergence_theorem}

If nonnegative measurable functions satisfy $f_n\uparrow f$ almost everywhere, then $\int f_n\uparrow\int f$, allowing the value infinity.

= Dominated convergence theorem
{parent=Measure theory}
{wiki}

If $f_n\to f$ almost everywhere and $|f_n|\leq g$ for one integrable function $g$, then $f$ is integrable and $\int|f_n-f|\to0$.

= Tonelli theorem
{parent=Measure theory}
{c}
{wiki=Fubini%27s_theorem\#Tonelli%27s_theorem}

Tonelli’s theorem permits interchange of integrals for a nonnegative measurable function, even before finiteness is known.

= Absolute continuity of measures
{parent=Measure theory}
{wiki=Absolute_continuity\#Absolute_continuity_of_measures}

A measure $\nu$ is absolutely continuous with respect to $\mu$ when every $\mu$-null set is also $\nu$-null.

= Lusin condition N
{parent=Absolute continuity of measures}
{c}

A function $f$ has Lusin's condition $(N)$ when it maps every <Lebesgue measure>[Lebesgue-null] set to a Lebesgue-null set.

= Image-length measure of a strictly increasing continuous function
{parent=Absolute continuity of measures}

Let $h:\mathbb R\to\mathbb R$ be <continuous function>[continuous], <strictly increasing function>[strictly increasing], and satisfy <Lusin condition N>. Then
$$
\nu(A)=\lambda(h(A))
$$
defines a <measure> on the <Lebesgue measurable set>[Lebesgue measurable sets], and $\nu\ll\lambda$. Moreover,
$$
\nu([a,b])=h(b)-h(a)
$$
for $a\leq b$.

= Uniform absolute continuity for a finite measure
{parent=Absolute continuity of measures}

If $\mu(\Omega)<\infty$, then $\mu\ll\nu$ exactly when for every $\varepsilon>0$ there is $\delta>0$ such that $\nu(A)<\delta$ implies $\mu(A)<\varepsilon$. For necessity, a contrary sequence with $\nu(A_n)<2^{-n}$ and $\mu(A_n)\geq\varepsilon$ has tail unions decreasing to a $\nu$-null limsup; continuity from above for finite $\mu$ contradicts absolute continuity.

= Mutually absolutely continuous measures
{parent=Absolute continuity of measures}

Two measures are mutually absolutely continuous when they have exactly the same null sets.

= Radon-Nikodym theorem
{parent=Absolute continuity of measures}
{c}
{wiki}

For sigma-finite measures, $\nu\ll\mu$ implies $\nu(A)=\int_A(d\nu/d\mu)d\mu$ for a nonnegative measurable density unique almost everywhere.

= Positive Radon-Nikodym derivative
{parent=Radon-Nikodym theorem}

Mutual absolute continuity is equivalent to the Radon--Nikodym derivative being finite and strictly positive almost everywhere.

= Dominating mixture measure
{parent=Absolute continuity of measures}

A countable family of probability measures is dominated by any mixture $\sum_nw_n\nu_n$ with every $w_n>0$ and $\sum_nw_n=1$.

= Lebesgue-Stieltjes measure
{title2=$\mu_F$}
{parent=Measure theory}
{c}
{wiki=Lebesgue%E2%80%93Stieltjes_integration}

An increasing right-continuous function $F:\mathbb R\to\mathbb R$ determines a measure by
$$
\mu_F((a,b])=F(b)-F(a).
$$
The <Radon-Nikodym theorem> and <Lebesgue differentiation theorem> identify the density of the absolutely continuous part of this measure with $F'$ almost everywhere.

= Lp inclusion on a finite measure space
{parent=Measure theory}
{wiki=Lp_space\#Embeddings}

On a finite measure space, $L^q\subseteq L^p$ for $0<p<q\leq\infty$.

= Power singularity integrability
{parent=Lp inclusion on a finite measure space}

On $(0,1)$, $x^{-a}$ is integrable exactly when $a<1$, providing sharp counterexamples between finite-measure $L^p$ spaces.

= Partial differential equation
{parent=Analysis}
{wiki}

A partial differential equation relates a multivariable function to its partial derivatives.

= Modified Helmholtz equation
{title2=$(\Delta-m^2)u=0$}
{parent=Partial differential equation}
{wiki=Helmholtz_equation}

The modified Helmholtz equation is
$$
\Delta u-m^2u=0.
$$
It reduces to the <Laplace equation> when $m=0$.

= Poisson equation
{parent=Partial differential equation}
{c}
{wiki}

= Poisson kernel for the upper half-space
{parent=Poisson equation}
{c}
{wiki}

The upper-half-space Poisson kernel gives the harmonic extension of boundary data.

= Poisson kernel
{synonym}

= Poisson integral
{parent=Poisson kernel for the upper half-space}
{c}
{wiki=Poisson_kernel}

The Poisson integral convolves boundary data with the <Poisson kernel> to produce a <harmonic function> in a half-space. In the upper half-plane,
$$
u(x,y)=\frac1\pi\int_{\mathbb R}\frac{y f(v)}{(x-v)^2+y^2}\,dv.
$$

= Diffusion equation
{parent=Partial differential equation}
{wiki}

A diffusion equation describes smoothing caused by a flux down spatial gradients.

= Fick's first law
{title2=$\mathbf J=-D\nabla C$}
{parent=Diffusion equation}
{c}
{wiki=Fick%27s_laws_of_diffusion}

Fick's first law says that diffusive flux points down the concentration gradient:
$$
\mathbf J=-D(C)\nabla C.
$$
Combining it with local conservation gives
$C_t=\nabla\cdot(D(C)\nabla C)$.

= Heat equation
{parent=Diffusion equation}
{wiki}

The heat equation
$$
u_t=D\nabla^2u,
\qquad D>0,
$$
models diffusion with constant diffusivity $D$.

= Sinusoidally forced heat equation on a half-line
{parent=Heat equation}

For $T_t=\kappa T_{xx}$ on $x>0$, with $T(x,0)=0$ and $T(0,t)=\sin\omega t$,
$$
\widetilde T(x,p)=\frac{\omega}{p^2+\omega^2}
\exp\left(-x\sqrt{\frac p\kappa}\right).
$$
Consequently the transform of $I(t)=\int_0^\infty T(x,t)\,dx$ is
$$
\widetilde I(p)=\frac{\omega\sqrt\kappa}
{\sqrt p\,(p^2+\omega^2)}.
$$

= Potential Burgers equation
{parent=Heat equation}

The nonlinear equation
$$
u_t=u_{xx}+u_x^2
$$
is the potential form of the viscous Burgers equation. The substitution $w=e^u$ converts it to the <heat equation> $w_t=w_{xx}$.

= Heat kernel
{parent=Heat equation}
{wiki}

For $u_t=\kappa u_{xx}$ on the real line,
$$
K_t(x)=\frac1{\sqrt{4\pi\kappa t}}e^{-x^2/(4\kappa t)}
$$
has total mass one and converges to the delta distribution as $t$ decreases to zero. The solution is $u(t)=K_t*u_0$.

= Gaussian approximate identity
{parent=Heat kernel}

The centered normal densities
$$
g_t(x)=\frac1{\sqrt{2\pi t}}e^{-x^2/(2t)}
$$
form an <approximate identity>: for every $f\in L^1(\mathbb R)$, $f*g_t\to f$ at almost every point where the <Lebesgue differentiation theorem> applies. Their Fourier transforms are $e^{-t\xi^2/2}$ in the angular-frequency convention.

= Heat-kernel solution
{parent=Heat kernel}

For initial data $u_0$ on the real line, Fourier transformation or convolution with the fundamental solution gives
$$
u(x,t)=\int_{-\infty}^{\infty}
\frac{e^{-(x-\xi)^2/(4Dt)}}{\sqrt{4\pi Dt}}
u_0(\xi)\,d\xi.
$$

= Heat equation with interval-indicator initial data
{parent=Heat-kernel solution}

For $u_t=Du_{xx}$ and $u(x,0)=\mathbf1_{[-a,a]}(x)$, convolution with the <heat kernel> gives
$$
u(x,t)=\frac12\left[
\operatorname{erf}\left(\frac{x+a}{2\sqrt{Dt}}\right)
-\operatorname{erf}\left(\frac{x-a}{2\sqrt{Dt}}\right)
\right].
$$

= Duhamel principle
{parent=Heat equation}
{c}
{wiki=Duhamel%27s_principle}

For a linear evolution equation with zero initial data, Duhamel's principle integrates the homogeneous propagator against the forcing time. For the forced heat equation,
$$
u(x,t)=\int_0^t\int_{-\infty}^{\infty}
K_{t-\tau}(x-\xi)f(\xi,\tau)\,d\xi\,d\tau.
$$

= Cancellation of two heat-kernel impulses
{parent=Duhamel principle}

On the real line, initial data $\delta(x-2\sqrt D)$ and a source $-A\delta(x+2\sqrt D)\delta(t-1)$ produce
$$
K_t(x-2\sqrt D)-AH(t-1)K_{t-1}(x+2\sqrt D).
$$
At $x=0,t=2$, the two terms cancel exactly for $A=\sqrt{e/2}$.

= Advection-diffusion equation
{parent=Diffusion equation}
{wiki}

The constant-coefficient one-dimensional advection-diffusion equation is
$$
u_t+a u_x=\kappa u_{xx}.
$$
Translation to coordinates moving at speed $a$ reduces it to the heat equation.

= Advection-diffusion heat-kernel solution
{parent=Advection-diffusion equation}

The Cauchy problem with initial value $u_0$ has
$$
u(t,x)=\int_{-\infty}^{\infty}K_t(x-at-y)u_0(y)\,dy.
$$

= Nonlinear diffusion equation
{parent=Diffusion equation}
{wiki=Porous_medium_equation}

A nonlinear diffusion equation lets diffusivity depend on the evolving field; the porous-medium equation is a standard example.

= Two-dimensional Barenblatt solution with diffusivity proportional to concentration
{c}
{parent=Nonlinear diffusion equation}

For $C_t=k\nabla\cdot(C\nabla C)$ with total mass $2\pi M$,
$$
C(r,t)=\sqrt{\frac{M}{kt}}
\left(\frac1{\sqrt2}-\frac18\frac{r^2}{\sqrt{Mkt}}\right)_+.
$$
Its compact support has radius $r_0(t)=(32Mkt)^{1/4}$.

= Linear-reaction time change for quadratic nonlinear diffusion
{parent=Two-dimensional Barenblatt solution with diffusivity proportional to concentration}

The substitution
$$
C(x,t)=e^{at}G(x,\tau(t)),
\qquad
\tau(t)=
\begin{cases}
(e^{at}-1)/a,&a\ne0,\\
t,&a=0,
\end{cases}
$$
reduces $C_t=k\nabla\cdot(C\nabla C)+aC$ to
$G_\tau=k\nabla\cdot(G\nabla G)$.

= Reaction-diffusion system
{parent=Diffusion equation}
{wiki=Reaction–diffusion_system}

A reaction-diffusion system combines local reaction kinetics with spatial diffusion,
$$
u_t=D\Delta u+f(u).
$$

= Morphogen reaction-diffusion equation
{parent=Reaction-diffusion system}
{wiki=Morphogen}

A morphogen reaction-diffusion equation models a spatial concentration by diffusion and local production or decay. Linearization at a homogeneous equilibrium $C=0$ replaces $f(C)$ by $f'(0)C$.

= Mixed Dirichlet-Neumann modes on an interval
{parent=Morphogen reaction-diffusion equation}

The eigenfunctions satisfying $X(0)=0$ and $X'(L)=0$ are
$$
X_n(x)=\sin(k_nx),
\qquad
k_n=\frac{(n+1/2)\pi}{L},quad n\geq0.
$$

= Critical length for a linearly growing morphogen
{parent=Mixed Dirichlet-Neumann modes on an interval}

For $C_t=DC_{xx}+aC$ with $a>0$, $C(0,t)=0$, and $C_x(L,t)=0$, every mode decays exactly when
$$
L<\frac\pi2\sqrt{\frac Da}.
$$

= Fast-inhibitor elimination in a reaction-diffusion system
{parent=Reaction-diffusion system}

If a fast inhibitor satisfies $0=v_{xx}+u-v$, then its spatial <Fourier transform> obeys
$$
\widehat v(k)=\frac{\widehat u(k)}{1+k^2}.
$$
Substitution produces a nonlocal scalar equation for the activator.

= Dispersion relation after fast-inhibitor elimination
{title2=$\sigma(k)$}
{parent=Fast-inhibitor elimination in a reaction-diffusion system}

For activator diffusion $D$, local linear decay $r$, and coupling $\rho(u-v)$, eliminating the fast inhibitor gives
$$
\sigma(k)=-Dk^2-r+\rho\frac{k^2}{1+k^2}.
$$
Its pattern-forming threshold is
$$
\rho_c=(\sqrt r+\sqrt D)^2,
\qquad
k_c=(r/D)^{1/4}.
$$

= Turing instability
{parent=Reaction-diffusion system}
{c}
{wiki=Turing_pattern}

A Turing, or diffusion-driven, instability occurs when a spatially homogeneous equilibrium is stable to homogeneous perturbations but unstable to a nonzero spatial Fourier mode after unequal diffusion is introduced.

= Two-species diffusion-driven instability criterion
{parent=Turing instability}

Let a stable two-species reaction equilibrium have Jacobian
$$
J=\begin{pmatrix}a&b\\c&d\end{pmatrix},
\qquad \operatorname{tr}J<0,
\qquad \det J>0,
$$
and diffusion matrix $\operatorname{diag}(D_u,D_v)$. A mode with $q=k^2$ has determinant
$$
\det(J-qD)=\det J-(D_va+D_ud)q+D_uD_vq^2.
$$
Since its trace decreases with $q$, diffusion-driven instability occurs exactly when
$$
D_va+D_ud>0,
\qquad
(D_va+D_ud)^2>4D_uD_v\det J.
$$

= Impossibility of a two-species Turing instability at equal diffusivities
{parent=Two-species diffusion-driven instability criterion}

When $D_u=D_v=D$, every spatial Fourier mode replaces the reaction Jacobian $J$ by $J-Dk^2I$. Each eigenvalue is shifted left by $Dk^2$, so diffusion cannot destabilize a stable homogeneous equilibrium.

= Near-unity diffusivity ratio for a Turing instability
{parent=Two-species diffusion-driven instability criterion}

For reaction Jacobian
$$
\begin{pmatrix}-1&-1\\1+\delta&1-\delta\end{pmatrix},
$$
the critical diffusivity ratio is
$$
d_c=
\left(
\frac{\sqrt{2\delta}+\sqrt{1+\delta}}{1-\delta}
\right)^2
=1+2\sqrt{2\delta}+O(\delta).
$$
It approaches one as the stable reaction system approaches marginality.

= Brusselator
{parent=Reaction-diffusion system}
{c}
{wiki}

The Brusselator is a two-species autocatalytic reaction model. In the parametrization
$$
f(u,v)=\alpha-(\beta+1)u+u^2v,
\qquad
g(u,v)=\beta u-u^2v,
$$
its positive homogeneous equilibrium is $(u_*,v_*)=(\alpha,\beta/\alpha)$.

= Brusselator trapping region
{parent=Brusselator}
{c}

For the spatially homogeneous Brusselator with parameters $r,s>0$, set $\alpha=r/(1+s)$. The quadrilateral
$$
\alpha\leq x,\qquad
0\leq y\leq\frac s\alpha,\qquad
x+y\leq r+\frac s\alpha
$$
is a <trapping region>. The four inward-pointing tests use $\dot x\geq0$ on $x=\alpha$, $\dot y\geq0$ on $y=0$, $\dot y\leq0$ on $y=s/\alpha$, and $\dot x+\dot y=r-x\leq0$ on the sloping edge.

= Brusselator periodic-orbit criterion
{parent=Brusselator}
{c}

The spatially homogeneous Brusselator has its unique positive equilibrium at $(r,s/r)$. Its Jacobian there has determinant $r^2$ and trace $s-1-r^2$. If $s-1>r^2$, the equilibrium is a repeller inside the <Brusselator trapping region>; the <Poincare-Bendixson theorem> then supplies a periodic orbit.

= Turing threshold of the Brusselator
{parent=Brusselator}
{c}

When $u$ and $v$ have diffusivities $D$ and $1$, respectively, and the homogeneous Brusselator equilibrium is stable, a <Turing instability> occurs precisely when
$$
\beta>(1+\alpha\sqrt D)^2.
$$
At onset,
$$
D_c=\frac{(\sqrt\beta-1)^2}{\alpha^2},
\qquad
k_*^2=\frac{\alpha^2}{\sqrt\beta-1}.
$$

= Similarity solution
{parent=Partial differential equation}
{wiki=Self-similarity\#Self-similar_solutions_of_differential_equations}

A similarity solution combines independent variables into scale-invariant coordinates and reduces a PDE to an ODE.

= Separation of variables
{parent=Partial differential equation}
{wiki}

Separation of variables seeks a product of one-variable factors, reducing a PDE to ordinary differential equations.

= Dirichlet problem on an annulus for one Fourier mode
{parent=Separation of variables}
{c}

The harmonic function on $a<r<b$ with boundary values $u(a,\theta)=0$ and $u(b,\theta)=\cos(n\theta)$ is
$$
u(r,\theta)=
\frac{b^n(r^{2n}-a^{2n})}{r^n(b^{2n}-a^{2n})}
\cos(n\theta).
$$

= Polynomial ansatz
{parent=Partial differential equation}
{wiki}

A polynomial ansatz determines an unknown polynomial solution by matching coefficients after applying the differential operator.

= Harmonic function
{parent=Partial differential equation}
{wiki}

A harmonic function satisfies Laplace's equation $\Delta u=0$.

= Harmonic
{synonym}

= Harmonic function as the real part of a holomorphic function
{parent=Harmonic function}

A real-valued function $u$ on a <simply connected domain> is <harmonic function>[harmonic] exactly when
$$
u=\operatorname{Re}f
$$
for some <holomorphic function> $f$. One construction observes that $u_x-iu_y$ is holomorphic and takes its <primitive of a holomorphic function on a simply connected domain>[holomorphic primitive].

= Subharmonic function
{parent=Harmonic function}
{wiki}

A twice differentiable function is strictly subharmonic where $\Delta u>0$. It cannot have an interior local maximum because the Hessian at such a maximum is negative semidefinite.

= Maximum principle for subharmonic functions
{parent=Subharmonic function}

A continuous <subharmonic function> on the closure of a bounded domain attains its maximum on the boundary. If it is twice differentiable, the strict condition $\Delta u>0$ rules out an interior maximum immediately because the <Hessian matrix> there would be negative semidefinite.

= Maximum principle for harmonic functions
{parent=Harmonic function}
{wiki=Maximum_principle}

A continuous function harmonic on a bounded domain attains its maximum and minimum on the boundary. Apply the strictly subharmonic result to $u+\epsilon|x|^2$ and let $\epsilon\downarrow0$, then apply the same argument to $-u$.

= Harmonic Liouville theorem
{parent=Harmonic function}
{c}
{wiki=Liouville%27s_theorem_(complex_analysis)\#Harmonic_functions}

A bounded <harmonic function> on the <complex plane>[plane] is constant. More generally, an entire harmonic function bounded on either side is constant: write it as $\operatorname{Re}f$ by <harmonic function as the real part of a holomorphic function>, then apply the <Liouville theorem> to $e^f$ or $e^{-f}$.

= Surjectivity of a nonconstant entire harmonic function
{parent=Harmonic Liouville theorem}

Every nonconstant harmonic function $u:\mathbb C\to\mathbb R$ is unbounded above and below by the <harmonic Liouville theorem>. The <intermediate value theorem> then gives $u(\mathbb C)=\mathbb R$.

= Scale-periodic harmonic function from an elliptic function
{parent=Harmonic function}

Let $\wp_\Lambda$ be the <Weierstrass elliptic function> for
$$
\Lambda=(\log q)\mathbb Z+2\pi i\mathbb Z,
\qquad q>1.
$$
Because $2\pi i$ is a <period lattice>[period], the formula
$$
u(z)=\operatorname{Re}\wp_\Lambda(\log z)
$$
is independent of the branch of the <complex logarithm>. It is a nonconstant harmonic function on
$$
\mathbb C^*\setminus\{q^m:m\in\mathbb Z\}
$$
and its other period gives $u(qz)=u(z)$.

= Harmonic polynomial
{parent=Harmonic function}
{wiki}

A harmonic polynomial is a polynomial annihilated by the Laplacian.

= Harmonic conjugate
{parent=Harmonic function}
{wiki}

A harmonic conjugate v makes u+iv holomorphic; it exists globally on simply connected domains.

= Local harmonic conjugate
{parent=Harmonic conjugate}

On a disc, the closed one-form $-u_y\,dx+u_x\,dy$ has a potential $v$ when $\Delta u=0$. The Cauchy--Riemann equations then make $u+iv$ holomorphic.

= Log modulus has no global harmonic conjugate on the punctured plane
{title2=$\log|z|$ on $\mathbb C^*$}
{parent=Harmonic conjugate}

The function $\log|z|$ is <harmonic function>[harmonic] on the <punctured complex plane>, but it is not globally the real part of a <holomorphic function> there. Such a function would have derivative $1/z$, whose integral around the unit circle is $2\pi i$, whereas the integral of a derivative around a closed curve is zero.

= Harmonic functions are smooth
{parent=Harmonic function}

Every $C^2$ harmonic function is locally the real part of a holomorphic function and is therefore infinitely differentiable.

= Harmonic functions are real analytic
{parent=Harmonic function}
{wiki=Harmonic_function\#Regularity}

Harmonic functions admit locally convergent power-series expansions.

= Unique continuation for harmonic functions
{parent=Harmonic functions are real analytic}

A harmonic function on a connected domain that vanishes on a nonempty open subset vanishes everywhere.

= Smooth bump function
{parent=Partial differential equation}
{wiki=Bump_function}

A smooth bump function is a smooth function with compact support. The flat exponential construction gives bumps supported in prescribed balls.

= Disjoint-support zero-product construction
{parent=Smooth bump function}

Two nonzero functions with disjoint supports have identically zero pointwise product. Separated continuous tent functions or smooth bumps provide examples.

= Laplace operator
{parent=Partial differential equation}
{c}
{wiki=Laplacian}

The Laplace operator is the divergence of the gradient, $\Delta=\nabla\cdot\nabla$.

= Laplace equation
{parent=Laplace operator}
{c}
{wiki}

Laplace's equation is
$$
\Delta u=0.
$$
Its solutions are <harmonic functions>.

= Green function of the Laplacian
{title2=$G(x,x')$}
{parent=Laplace equation}
{c}
{wiki=Green%27s_function}

In three dimensions,
$$
-\nabla^2\frac1{4\pi|x-x'|}=\delta^{(3)}(x-x').
$$
Consequently the decaying solution of $-\nabla^2u=f$ in free space is
$$
u(x)=\frac1{4\pi}\int_{\mathbb R^3}\frac{f(x')}{|x-x'|}\,d^3x'.
$$

= Laplace's equation
{c}
{synonym}

= Laplace equation in polar coordinates
{parent=Laplace equation}
{c}

In plane <polar coordinates>, <Laplace's equation> is
$$
\frac1r\frac{\partial}{\partial r}
\left(r\frac{\partial\phi}{\partial r}\right)
+\frac1{r^2}\frac{\partial^2\phi}{\partial\theta^2}=0.
$$
<Separation of variables> gives radial powers $r^n,r^{-n}$ for positive angular modes and $1,\log r$ for the zero mode.

= Radial Laplacian
{parent=Laplace operator}
{wiki}

For radial $u(r)$ in $n$ dimensions, $\Delta u=u\prime\prime+(n-1)u\prime/r$.

= Laplacian in spherical coordinates
{parent=Laplace operator}
{c}
{wiki}

The spherical-coordinate Laplacian splits into a radial operator and the sphere’s angular Laplacian divided by $r^2$.

= Axisymmetric harmonic function
{parent=Laplacian in spherical coordinates}
{wiki}

Axisymmetric harmonic functions separate into radial powers times Legendre polynomials.

= Laplacian eigenfunction
{parent=Laplace operator}
{c}
{wiki}

A Laplacian eigenfunction satisfies $-\Delta u=\lambda u$ together with specified boundary conditions.

= Dirichlet Laplacian eigenfunction
{parent=Laplacian eigenfunction}
{c}
{wiki}

A Dirichlet Laplacian eigenfunction vanishes on the boundary and satisfies $-\Delta u=\lambda u$.

= Wave equation
{parent=Partial differential equation}
{wiki}

The wave equation models finite-speed propagation and second-order oscillation.

= Elastic wave
{parent=Wave equation}
{wiki=Elastic_wave}

Small displacements in an isotropic elastic solid satisfy a vector wave equation governed by density and the two Lamé moduli.

= Helmholtz separation of elastic waves
{parent=Elastic wave}
{c}

Divergence of the elastic equation isolates dilatational motion, while curl isolates rotational motion, producing P and S wave equations with different speeds.

= P-wave
{parent=Elastic wave}
{c}
{wiki=P_wave}

A P-wave is longitudinal, carries nonzero dilatation, and has speed $c_P=\sqrt{(\lambda+2\mu)/\rho}$ in an isotropic solid.

= Longitudinal polarization
{parent=P-wave}

In a longitudinal plane wave, displacement is parallel to the wavevector and the curl vanishes.

= S-wave
{parent=Elastic wave}
{c}
{wiki=S_wave}

An S-wave is transverse, carries rotation without dilatation, and has speed $c_S=\sqrt{\mu/\rho}$.

= Transverse polarization
{parent=S-wave}

In a transverse plane wave, displacement is perpendicular to the wavevector and its divergence vanishes.

= SV-wave
{c}
{parent=S-wave}
{wiki=SV_wave}

An SV-wave is polarized in the vertical plane containing its wavevector and perpendicular to that wavevector.

= SH-wave
{c}
{parent=S-wave}
{wiki=SH_wave}

An SH-wave is polarized horizontally and perpendicular to the vertical propagation plane.

= Guided SH modes between rigid and free planes
{parent=SH-wave}

For a layer $0<z<h$ with rigid lower boundary and traction-free upper boundary, an SH mode
$u_y=Y(z)e^{i(\kappa x-\omega t)}$ has
$$
Y(z)=\sin(q_nz),\qquad
q_n=\frac{(2n+1)\pi}{2h},
$$
and dispersion
$$
\omega^2=c_s^2(\kappa^2+q_n^2).
$$
Its cutoff is $\omega_n=c_sq_n$, its phase speed is $\omega/\kappa$, and its group speed is $c_s^2\kappa/\omega$.

= Elastic-interface boundary conditions
{parent=Elastic wave}

At a perfectly bonded interface, all displacement and traction components are continuous.

= P-SV mode conversion at a solid interface
{parent=Elastic-interface boundary conditions}
{c}

An obliquely incident in-plane P-wave generally produces reflected and transmitted P and SV waves; SH polarization decouples.

= Snell law for elastic waves
{parent=Elastic-interface boundary conditions}
{c}

Phase matching fixes a common frequency and tangential wavenumber for every incident, reflected, and transmitted elastic mode.

= Acoustic reflection and transmission at an interface
{parent=Elastic wave}

For two inviscid media, continuity of normal displacement and normal stress determines the reflected and transmitted longitudinal amplitudes.

= Equal displacement-amplitude condition at an acoustic interface
{parent=Acoustic reflection and transmission at an interface}

Equating the absolute reflected and transmitted displacement amplitudes, then using acoustic Snell's law, gives an algebraic condition on incidence angle and the two moduli-over-speed impedances.

= Wave equation on a string
{parent=Wave equation}
{wiki}

A stretched uniform string obeys y_tt=c^2 y_xx with wave speed equal to the square root of tension over density.

= Linearly damped string
{parent=Wave equation on a string}

With fixed endpoints, $y_{tt}-y_{xx}+y_t=0$ separates into sine modes whose amplitudes satisfy
$$
T_n''+T_n'+n^2\pi^2T_n=0.
$$

= Critical damping
{parent=Linearly damped string}
{wiki=Damping#Critical_damping}

Critical damping occurs when a second-order mode has a repeated real characteristic root. It returns to equilibrium without oscillating and as quickly as possible among nonoscillatory regimes.

= Separated solution of the damped string equation
{parent=Linearly damped string}

For initial sine coefficient $a_n$ and zero initial velocity, put $\omega_n^2=n^2\pi^2-1/4$. The modal amplitude is
$$
T_n(t)=a_ne^{-t/2}
\left(\cos(\omega_nt)+\frac{\sin(\omega_nt)}{2\omega_n}\right).
$$

= Energy dissipation identity for a linearly damped string
{parent=Linearly damped string}

For fixed endpoints and
$$
E(t)=\frac12\int_0^1(y_t^2+y_x^2)\,dx,
$$
integration by parts and the damped wave equation give
$$
E'(t)=-\int_0^1y_t^2\,dx\leq0.
$$

= Normal mode
{parent=Wave equation}
{wiki}

A normal mode oscillates at one frequency with a fixed spatial eigenfunction.

= Modal energy distribution
{parent=Normal mode}
{wiki}

Orthogonality separates total quadratic wave energy into a sum of independent modal energies.

= Node of a standing wave
{parent=Normal mode}
{wiki}

A node is a point where a standing-wave eigenfunction vanishes at all times.

= Fourier transform method for the wave equation
{parent=Wave equation}
{c}

Fourier transformation in space turns $u_{tt}=c^2u_{xx}$ into the independent oscillators $\widehat u_{tt}+c^2k^2\widehat u=0$.

= D'Alembert formula
{parent=Wave equation}
{c}
{wiki=D%27Alembert%27s_formula}

The Cauchy problem $u_{tt}=c^2u_{xx}$ has
$$u(x,t)=\frac{f(x-ct)+f(x+ct)}2+\frac1{2c}\int_{x-ct}^{x+ct}g(y)dy.$$

= D'Alembert formula with initial velocity
{parent=D'Alembert formula}
{c}

For $u_{tt}-u_{xx}=0$ with $u(0,x)=0$ and $u_t(0,x)=f(x)$,
$$
u(t,x)=\frac12\int_{x-t}^{x+t}f(y)\,dy.
$$

= Central zero region for odd compactly supported initial velocity
{parent=D'Alembert formula with initial velocity}

If $u(x,0)=0$, $u_t(x,0)=g(x)$, and the odd function $g$ is supported on $[-a,a]$, then
$$
u(x,t)=0
\qquad\text{whenever }t>a\text{ and }|x|\leq t-a.
$$
The interval $[x-t,x+t]$ then contains the entire support, whose integral vanishes by oddness.

= Wavenumber
{title2=$k$}
{parent=Wave equation}
{wiki}

The wavenumber is the spatial angular frequency of a monochromatic wave. A factor $e^{ikx}$ has wavelength $2\pi/|k|$.

= Dispersion relation
{title2=$\omega(k)$}
{parent=Wave equation}
{wiki}

A dispersion relation gives angular frequency $\omega$ as a function of wavenumber $k$ for monochromatic waves admitted by a linear wave equation.

= Growth rate
{title2=$\operatorname{Re}\sigma$}
{parent=Dispersion relation}

For a normal mode proportional to $e^{\sigma t}$, the real part $\operatorname{Re}\sigma$ is its exponential growth rate. A positive growth rate signals linear instability.

= Phase velocity and group velocity
{parent=Dispersion relation}
{wiki=Group_velocity}

For a dispersion relation $\omega(k)$, the phase velocity is $c_p=\omega/k$ and the group velocity is $c_g=d\omega/dk$. For the Klein-Gordon relation at $k>0$,
$$
c_p=c\frac{\sqrt{k^2+A^2}}{k}>c,
\qquad
c_g=c\frac{k}{\sqrt{k^2+A^2}}<c,
\qquad
c_pc_g=c^2.
$$

= Wave crest
{parent=Phase velocity and group velocity}
{wiki=Crest_(physics)}

A wave crest is a point of constant phase. For a monochromatic wave it propagates at the <phase velocity and group velocity>[phase velocity].

= Wave packet
{parent=Phase velocity and group velocity}
{wiki}

A wave packet is a localized superposition of nearby wavenumbers. Its envelope propagates at the <phase velocity and group velocity>[group velocity] to leading order.

= Ninth-order dispersive advection equation
{parent=Phase velocity and group velocity}

For
$$
\phi_t+U\phi_x+\frac19\phi_{xxxxxxxxx}=0,
$$
the dispersion relation is $\omega=Uk+k^9/9$, so $c_p=U+k^8/9$ and $c_g=U+k^8$.

= Klein-Gordon equation
{parent=Wave equation}
{c}
{wiki=Klein%E2%80%93Gordon_equation}

The one-dimensional Klein-Gordon equation
$$
\phi_{tt}-c^2\phi_{xx}+A^2c^2\phi=0
$$
has positive-frequency dispersion relation $\omega(k)=c\sqrt{k^2+A^2}$.

= Stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray
{parent=Klein-Gordon equation}

For zero initial velocity and Fourier amplitude $a(k)$, observation along $x=Vt$ with $0\leq V<c$ selects the two stationary wavenumbers $\pm k_0$, where
$$
k_0=\frac{AV}{\sqrt{c^2-V^2}}.
$$
Writing $s=\sqrt{c^2-V^2}$, the leading oscillation has angular frequency $As$ and amplitude proportional to $t^{-1/2}$.

= Upward zero crossings of an oscillatory stationary-phase tail
{parent=Stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray}

If the leading asymptotic is $Ct^{-1/2}\cos(\Omega t+\gamma)$ with $C>0$, its upward zero crossings satisfy
$$
\Omega t+\gamma=\frac{3\pi}{2}+2\pi n
$$
to leading order.

= Green second identity
{parent=Partial differential equation}
{c}
{wiki}

Green’s second identity is the divergence theorem applied to phi grad psi minus psi grad phi.

= Green's third identity
{c}
{parent=Green second identity}
{wiki=Green%27s_identities}

Green's third identity represents a function inside a domain through its boundary values, normal derivatives, and a volume term involving its Laplacian.

= Green's first identity
{c}
{parent=Partial differential equation}
{wiki=Green%27s_identities}

For smooth functions $u,v$ on a region $V$,
$$
\int_V\left(u\Delta v+\nabla u\cdot\nabla v\right)dV
=\int_{\partial V}u\frac{\partial v}{\partial n}\,dS.
$$

= Lie point symmetry
{parent=Partial differential equation}
{c}
{wiki=Lie_point_symmetry}

A Lie point symmetry is a continuous transformation of independent and dependent variables that maps solutions of a differential equation to solutions.

= Infinitesimal generator of a Lie point symmetry
{parent=Lie point symmetry}

A one-parameter point transformation has infinitesimal generator
$$
V=\sum_i\xi^i(x,u)\partial_{x_i}+\eta(x,u)\partial_u.
$$
Its flow recovers the finite transformations.

= Prolongation of a Lie point symmetry
{parent=Lie point symmetry}

The prolongation of a point-symmetry generator is its induced vector field on derivatives of the dependent variable. A generator is a symmetry of a differential equation $F=0$ when its required prolongation sends $F$ to zero on the solution manifold.

= Second prolongation of a Lie point symmetry
{parent=Prolongation of a Lie point symmetry}

For $V=\xi^i\partial_{x_i}+\eta\partial_u$, the second prolongation adds coefficients for $u_i$ and $u_{ij}$ obtained by total differentiation:
$$
\eta_i=D_i\eta-u_jD_i\xi^j,
\qquad
\eta_{ij}=D_j\eta_i-u_{ik}D_j\xi^k.
$$

= Scaling symmetry of a partial differential equation
{parent=Lie point symmetry}

A scaling is a symmetry when every term of the equation transforms with the same weight.

= Simultaneous spacetime scaling symmetry of the wave equation
{parent=Scaling symmetry of a partial differential equation}

The flow of $V=t\partial_t+x\partial_x$ is $(t,x,u)\mapsto(e^\epsilon t,e^\epsilon x,u)$. Its second prolongation scales both $u_{tt}$ and $u_{xx}$ with weight minus two, so it preserves $u_{tt}-u_{xx}=0$.

= Symmetry reduction of a partial differential equation
{parent=Lie point symmetry}

An invariant of a one-parameter symmetry group becomes a similarity variable that reduces a PDE to an ODE.

= Group-invariant solution
{parent=Symmetry reduction of a partial differential equation}

A group-invariant solution is constant along the prolonged symmetry orbits. For a scaling of two independent variables with fixed dependent variable, it depends only on a ratio such as $x/t$.

= Projective Lie symmetry of the potential Burgers equation
{parent=Symmetry reduction of a partial differential equation}

The vector field
$$
V=4t^2\partial_t+4tx\partial_x-(x^2+2t)\partial_u
$$
generates the local transformations
$$
T=\frac{t}{1-4\epsilon t},
\quad
X=\frac{x}{1-4\epsilon t},
\quad
U=u-\frac{\epsilon x^2}{1-4\epsilon t}
+\frac12\log(1-4\epsilon t).
$$
Its second prolongation sends $u_t-u_{xx}-u_x^2$ to $-8t$ times that expression, so it is a <Lie point symmetry> of the <potential Burgers equation>.

= Similarity variable
{parent=Symmetry reduction of a partial differential equation}

A similarity variable is constant along symmetry-group orbits and parametrizes group-invariant solutions.

= Distribution theory
{parent=Analysis}
{wiki=Distribution_(mathematics)}

Distribution theory extends functions to continuous linear functionals on test functions, allowing generalized derivatives and point sources.

= Distributional identity
{parent=Distribution theory}

A distributional identity is an equality that holds after both sides act on every smooth compactly supported test function.

= Dirac delta function
{parent=Distribution theory}
{c}
{wiki}

The Dirac delta is the distribution defined by $\langle\delta,\varphi\rangle=\varphi(0)$.

= Distributional derivative of the Heaviside step function
{parent=Dirac delta function}

For every test function $\varphi$,
$$
\langle H',\varphi\rangle=-\int_0^\infty\varphi'(t)dt=\varphi(0),
$$
so $H'=\delta$ as distributions.

= Dirac delta scaling
{parent=Dirac delta function}
{c}

For nonzero $a$, $\delta(ax)=\delta(x)/|a|$ in the distributional sense.

= Calculus of variations
{parent=Analysis}
{wiki}

Calculus of variations finds <stationary points> of <functionals>, often among <functions> or <regular curves>.

= Functional
{parent=Calculus of variations}
{wiki=Functional_(mathematics)}

A functional is a <function> whose inputs are themselves functions, curves, or other elements of a function space.

= Energy functional
{parent=Functional}
{wiki=Dirichlet%27s_principle}

An energy functional assigns an integral energy to a function. For the modified Helmholtz equation, a natural example is
$$
E[u]=\int_V\left(|\nabla u|^2+m^2u^2\right)dV.
$$

= Lagrangian
{parent=Functional}
{wiki=Lagrangian_(field_theory)}

A Lagrangian is a function or density whose stationary integral encodes an extremization problem. In <Lagrangian mechanics> its action gives the equations of motion; in geometry it describes geodesics and minimal surfaces.

= Lagrangian function in constrained optimization
{parent=Lagrangian}
{wiki=Lagrange_multiplier\#The_Lagrangian}

For an objective $f(x)$ with equality constraints $g_i(x)=0$, the Lagrangian function is
$$
L(x,\lambda)=f(x)+\sum_i\lambda_i g_i(x).
$$

= Variation
{parent=Functional}
{wiki=Calculus_of_variations}

A variation of a function $y$ is a one-parameter family $y+\varepsilon\eta$, where $\eta$ satisfies the required <boundary conditions>.

= First variation
{parent=Variation}
{wiki=First_variation}

The first variation of a <functional> $L$ at $y$ in the direction $\eta$ is
$$
\delta L[y;\eta]
=\left.\frac d{d\varepsilon}L[y+\varepsilon\eta]\right|_{\varepsilon=0}.
$$

= Fundamental lemma of the calculus of variations
{parent=First variation}
{wiki}

If a continuous function $f$ satisfies
$$
\int_a^b f(x)\eta(x)\,dx=0
$$
for every smooth compactly supported test function $\eta$, then $f=0$ on $(a,b)$.

= Stationary point
{parent=Calculus of variations}
{wiki=Stationary_point}

A point is stationary for a <differentiable function> or <functional> when its first <derivative> or <first variation> vanishes in every admissible direction.

= Dirichlet principle
{parent=Calculus of variations}
{c}
{wiki}

Among functions with fixed boundary values, the solution of $\nabla\cdot(\kappa\nabla\phi)=0$ with $\kappa>0$ uniquely minimizes the weighted energy
$$
\int\kappa|\nabla\phi|^2.
$$

= Optical ray in cylindrical coordinates
{parent=Calculus of variations}
{wiki}

Fermat-type optical paths in cylindrical coordinates extremize refractive index times Euclidean arclength.

= Helical extremal
{parent=Optical ray in cylindrical coordinates}
{wiki}

A helical extremal has constant radius and a polar angle linear in axial position.

= Noether theorem
{parent=Calculus of variations}
{c}
{wiki}

Noether’s theorem associates a conserved quantity to each continuous variational symmetry.

= Second variation
{parent=Calculus of variations}
{wiki=Second_variation}

The second variation is the quadratic term in a functional's expansion along $u+\varepsilon\eta$; positivity on admissible variations is a local-minimum test.

= Jacobi equation
{parent=Second variation}
{c}
{wiki=Jacobi_field}

The Jacobi equation is the linearization of an <Euler-Lagrange equation> about a stationary path. Its solutions describe infinitesimal one-parameter families of stationary paths.

= Conjugate point
{parent=Second variation}
{wiki=Conjugate_points}

A conjugate point marks a nonzero endpoint-vanishing Jacobi variation and the loss of positive definiteness of the second variation.

= Wirtinger inequality
{parent=Second variation}
{c}
{wiki}

For $f(0)=f(T)=0$,
$$\int_0^T|f'|^2dt\geq\frac{\pi^2}{T^2}\int_0^T|f|^2dt,$$
with equality for a multiple of $\sin(\pi t/T)$.

= Periodic Wirtinger inequality
{parent=Wirtinger inequality}

If a continuously differentiable $L$-periodic function $f$ has mean zero, then
$$
\int_0^L f(s)^2\,ds
\leq\frac{L^2}{4\pi^2}\int_0^L f'(s)^2\,ds.
$$
Equality holds exactly for $f(s)=A\cos(2\pi s/L)+B\sin(2\pi s/L)$.

= Fixed-point theorem
{parent=Analysis}
{wiki=Fixed-point_theorem}

A fixed-point theorem gives conditions under which a map $T$ has a point satisfying $Tx=x$.

= Knaster-Tarski theorem
{c}
{parent=Fixed-point theorem}
{wiki=Knaster%E2%80%93Tarski_theorem}

Every monotone self-map of a complete lattice has a complete lattice of fixed points. Its least fixed point is the meet of its prefixed points, and its greatest fixed point is the join of its postfixed points.

= Fixed-point property
{parent=Fixed-point theorem}
{wiki=Fixed-point_property}

A topological space has the fixed-point property when every continuous self-map has a fixed point.

= Homeomorphism invariance of the fixed-point property
{parent=Fixed-point property}

If $h:A\to B$ is a <homeomorphism> and $A$ has the fixed-point property, then a continuous $f:B\to B$ induces $h^{-1}fh:A\to A$. A fixed point of the conjugate map is carried by $h$ to a fixed point of $f$.

= Fixed-point property of a closed interval
{parent=Fixed-point property}

For continuous $f:[0,1]\to[0,1]$, the function $f(x)-x$ is nonnegative at zero and nonpositive at one, so the <intermediate value theorem> gives a fixed point.

= Contraction mapping theorem
{parent=Fixed-point theorem}
{wiki=Banach_fixed-point_theorem}

A contraction of a nonempty complete metric space has one fixed point, and every orbit converges to it geometrically.

= Continuous dependence of the fixed point of a uniform contraction
{parent=Contraction mapping theorem}

Suppose $T_\lambda$ are contractions of one complete metric space with a common constant $k<1$, and $T_\lambda x$ depends continuously on $\lambda$ for each fixed $x$. Their fixed points $x_\lambda$ depend continuously on $\lambda$, because for a fixed parameter $\lambda_0$,
$$
d(x_\lambda,x_{\lambda_0})
\leq\frac{d(T_\lambda x_{\lambda_0},T_{\lambda_0}x_{\lambda_0})}{1-k}.
$$

= Iterated contraction
{parent=Contraction mapping theorem}

If some iterate $T^n$ is a contraction on a complete metric space, then $T$ itself has exactly one fixed point: uniqueness for $T^n$ forces $T$ to fix its fixed point.

= Local contraction proof for Newton iteration
{parent=Contraction mapping theorem}

For the Newton map $g(x)=x-f(x)/f'(x)$ near a simple root $r$,
$$
g'(x)=\frac{f(x)f''(x)}{f'(x)^2},
\qquad g'(r)=0.
$$
Bounds $|f'|\geq\delta$ and $|f''|\leq M$ make $|g'|<1$ on a sufficiently small closed interval around $r$. The contraction mapping theorem then gives the unique local fixed point.

= Uniform approximation
{parent=Analysis}
{wiki=Uniform_approximation}

Uniform approximation minimizes the <supremum norm> of the difference between a target <function> and an approximant.

= Best uniform approximation
{parent=Uniform approximation}

Given a family $A$ of approximating <functions>, a best uniform approximation to $f$ is an element $p\in A$ satisfying
$$
\lVert f-p\rVert_\infty=\inf_{q\in A}\lVert f-q\rVert_\infty.
$$

= Chebyshev alternation theorem
{parent=Uniform approximation}
{c}
{wiki=Equioscillation_theorem}

A degree-at-most-$m$ polynomial is a best uniform approximation precisely when its error attains alternating extrema of equal magnitude at at least $m+2$ ordered points.