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Every continuous self-map of a closed disc has a fixed point.
If a polynomial equation can be rearranged as with a continuous mapping a closed disk into itself, Brouwer's theorem guarantees a root in that disk.
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.
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 closed disc retracts continuously onto its boundary. This is equivalent to Brouwer fixed-point theorem by the standard ray construction and the antipodal map.
A metric space is a set equipped with a nonnegative symmetric distance satisfying definiteness and the triangle inequality.
A sequence converges to in a metric space when , equivalently when every neighbourhood of contains all sufficiently late terms.
A map between metric spaces is continuous exactly when always implies . If an inverse image of an open set were not open, points outside it could be chosen within distance of a point inside it, proving the converse by contradiction.
Every sequence in a compact metric space has a convergent subsequence. Conversely, sequential compactness and compactness are equivalent for metric spaces.
Every metric space is normal. For disjoint closed sets , the functionis continuous, equals zero on , and equals one on ; inverse images of disjoint neighbourhoods of zero and one separate the sets.
A metric space is compact exactly when every continuous function is bounded. For the converse, a noncompact metric space contains a closed discrete sequence; the unbounded function assigning its th point the value extends to by the Tietze extension theorem.
A metric space is complete when every Cauchy sequence converges to a point of the space.
A metric satisfies . A norm satisfies the corresponding inequality .
The Euclidean distance between is
The diameter of a subset of a metric space is
Uniform continuity requires one input tolerance to work at every point of the domain for a chosen output tolerance.
Every continuous map from a compact metric space to a metric space is uniformly continuous.
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.
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.
On , the usual metric and induce the same topology, but their ratio along tends to zero. Thus no positive global lower comparison constant exists.
Topological equivalence does not determine uniform convergence. For the usual and arctangent metrics on , maps that change the value to at a single domain point have arctangent sup-error tending to zero while their usual sup-error diverges.
For a metric space, is -Lipschitz and vanishes on the closure of .
A point of a metric space is isolated when some open ball about contains no other point of the space, equivalently when the singleton is open.
A metric space is complete when every Cauchy sequence converges to a point of the space.
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.
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 be nonempty and open and let be open dense sets. Successively choose closed ballsstarting with a ball contained in . The centres form a Cauchy sequence. Its limit belongs to every closed ball, and hence to .
Let be a nonempty closed, convex, symmetric subset of a Banach space and suppose . Baire's theorem puts a ball inside some . Symmetry also puts there, and convexity puts the midpoint of and , namely , in whenever . Hence .
Convexity is essential. In , letThis set is closed, symmetric, and has gaps arbitrarily near zero. Given , choose so small-scale that the interval
has length at least one; it contains an integer , and then . Thus , although is not a neighbourhood of zero.
has length at least one; it contains an integer , and then . Thus , although is not a neighbourhood of zero.
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.
A subset 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, or first-category, set is a countable union of nowhere dense sets.
In 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.
Let and suppose that for every there is an such that . Then is a polynomial.
Set and . The sets are closed, , and Baire's theorem on each interval shows that is dense. On every connected component of , compactness and the increasing cover by the show that one derivative vanishes locally with a uniform finite order, so agrees there with one polynomial.
If 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 gives an interval and an index for which is nonempty and contained in . Because has no isolated points, difference quotients show that every derivative of order at least vanishes on . Each polynomial component of meeting has an endpoint in , so its degree is less than . Thus throughout , contradicting . Hence , and its sole connected component carries one polynomial.
The space becomes a complete metric space underConvergence in this metric is exactly uniform convergence of every derivative separately.
If is Cauchy in the smooth-function metric, each derivative sequence converges uniformly to a continuous function . Passing to the limit inshows that . Hence is smooth and in the metric.
Outside a meagre set in , every rational and every positive integer admit such thatFor fixed , the union over of these strict-inequality sets is open. It is dense because a perturbation can make the th derivative arbitrarily large while keeping any prescribed finite collection of lower derivatives arbitrarily small. The Baire category theorem applied over the countable pairs gives the claim.
The generic derivative bound implies that at every rational ,The Cauchy-Hadamard theorem therefore gives radius zero to the Taylor series at every such . If a Taylor series represented on a neighborhood of any point, that neighborhood would contain a rational point at which was real analytic, a contradiction.
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.
For nonempty bounded subsets of a metric space,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.
The map 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.
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.
A family of maps between metric spaces is equicontinuous when, near each point, one input tolerance controls the output variation uniformly for every . Uniform equicontinuity uses one input tolerance over the whole domain.
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 .
If successive subsequences converge uniformly on an exhaustion , the diagonal sequence converges uniformly on every fixed , hence locally uniformly on their union.
For a compactly supported nonzero function , the translates converge uniformly to zero on every compact subset of , while prevents uniform convergence on all of .
An integral operator maps a function to a function of the formA bounded continuous kernel on a compact domain defines a bounded operator in the uniform norm.
A normed vector space is a vector space equipped with a norm, whose induced metric is .
For , the L1 norm is
The Euclidean norm of is
The closed Euclidean ball of radius about is ; replacing by gives the open ball.
The supremum norm of a bounded real- or complex-valued function on a set isFor a continuous function on a compact space, the extreme value theorem makes this supremum a maximum.
A surjective isometry between normed vector spaces preserves every distance:
Every surjective isometry between real normed vector spaces is affine. In particular, if , then is real-linear.
For in a real normed space, start with the points at distance from both endpoints and repeatedly retain those within half the previous diameter of every retained point. Reflection about 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.
Surjectivity in the Mazur-Ulam theorem is essential. The mappreserves all distances and fixes zero, but it is not linear and does not preserve every midpoint.
A Banach space is a normed vector space that is complete in its norm metric.
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.
A point-separating real subalgebra of that contains the constants is uniformly dense when 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.
Every continuous real-valued function on a compact interval is a uniform limit of polynomials.
For , its tensor-product Bernstein polynomial is the expectation of for independent binomial variables . Uniform continuity and concentration of the binomial variables prove uniform convergence to .
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.
A bounded bijective linear map between Banach spaces has a bounded inverse. This follows because an open bijection has continuous inverse.
A diagonal operator on acts coordinatewise as . It is bounded when is bounded, while its inverse on its image is bounded only when the nonzero are bounded away from zero.
The injective diagonal map on is bounded, but its inverse on the range is unbounded; the range is correspondingly not closed.
The graph of is the linear subspace of .
A linear map between Banach spaces is continuous if and only if its graph is closed.
For a closed graph, the projection is a bounded bijection between Banach spaces. Its bounded inverse followed by projection to is .
If bounded is injective and , then has closed graph: and imply and hence .
On incomplete with the norm, let and let be inclusion into . Their images agree, but is unbounded.
Let be a real Banach space and be linear with for every . If and in , positivity of and passage to the limit givefor every real and every . Taking small of either sign forces . Thus the graph of is closed, and the closed graph theorem makes continuous.
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.
If is invertible and , then is invertible by the Neumann series. Similar iterative corrections prove stability of surjectivity.
A linear subspace is dense exactly when the only continuous functional vanishing on it is zero, by Hahn-Banach separation.
A Hilbert space is a complete inner-product space.
Every nonempty closed convex subset of a Hilbert space contains a unique point nearest to each . A minimizing sequence is Cauchy by the parallelogram identity, and uniqueness follows by applying the same identity to two minimizers and their midpoint.
For a closed subspace of a Hilbert space,The closest point gives the first component, and differentiating at zero shows that the remainder is orthogonal to every .
Every bounded linear functional on a Hilbert space is inner product with one unique vector, with equal norms.
For , the adjoint is the unique bounded operator satisfyingRiesz representation applied for each proves existence; in an orthonormal basis its matrix is the conjugate transpose of the matrix of .
A bounded operator is Hermitian, or self-adjoint, when .
For a subspace of a finite-dimensional inner-product space,This follows directly by moving across the inner product and using .
An operator is normal when . Equivalently,for every ; the converse follows by polarizing the quadratic form of the self-adjoint commutator .
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.
A sequence converges weakly to when for every in the Hilbert space.
For a Hilbertian basis , a sequence converges weakly if and only if is norm bounded and every coordinate sequence converges.
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.
If in a Hilbert space, thenFor , take the inner product with and apply Cauchy-Schwarz before passing to the lower limit.
In a Hilbert space, weak convergence together with implies strong convergence .
Suppose in a separable Hilbert space with Hilbertian basis . Then in norm exactly whenThe condition prevents norm from escaping to successively higher basis coordinates.
Every orthonormal sequence in a Hilbert space converges weakly to zero, by Bessel inequality, but no subsequence converges strongly because distinct terms remain distance apart.
If converges weakly to in a Banach space, there are convex combinations of each tail that converge in norm to .
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.
An orthonormal sequence satisfies for and .
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.
For a Hilbertian basis ,
For an orthonormal sequence in a Hilbert space,
A bounded operator is compact when it maps the unit ball to a relatively compact set.
An operator on a Hilbert space is Hilbert--Schmidt whenfor one, equivalently every, Hilbertian basis. Every Hilbert--Schmidt operator is compact.
A finite-rank operator has finite-dimensional image and is compact.
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.
Let project onto its first coordinates. Although only strongly on the whole unit ball, convergence is uniform on every compact subset. Hence compact satisfiesand each has finite rank.
A real-valued functional is sublinear when
and for every .
and for every .
Every bounded linear functional on a linear subspace of a real normed space extends to the whole space without increasing its norm.
For a normed space , the map defined byis a linear isometry. The easy inequality is ; Hahn--Banach extends the norm-one functional on that takes to , proving the reverse inequality.
Point evaluation at zero is bounded on with the essential-supremum norm, and Hahn--Banach extends it to a bounded functional on . No represents this extension: continuous functions supported in shrinking neighbourhoods of zero have value one at zero while their integrals against tend to zero. Hence is strictly larger than .
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.
The closed unit ball of a dual Banach space is compact in the weak-star topology.
A bounded coercive bilinear form on a Hilbert space represents every bounded linear functional uniquely: for each , there is a unique with for every .
The solution operator is compact on every bounded open set because it maps boundedly into and the Rellich-Kondrashov compactness theorem for H01 embeds that space compactly into .
A sequence bounded in whose members are supported in one bounded set has a subsequence converging strongly in . Apply Rellich compactness simultaneously to the functions and their first derivatives.
For , restriction to the coordinate hyperplane extends uniquely to a bounded linear mapIn Fourier variables, Cauchy--Schwarz bounds the integral over the normal frequency because is integrable exactly when .
For , no bounded map from to can agree with restriction on continuous functions. Indeed,has a fixed nonzero trace while .
If , then embeds continuously into bounded continuous functions.
For every bounded open , the inclusion is compact.
By the definition of as the closure of compactly supported smooth functions, extension by zero maps it continuously into without creating a boundary distribution.
For , Plancherel gives
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.
On bounded with , weak convergence in makes the Dirichlet term lower semicontinuous and, by Rellich compactness, makes the potential term continuous. Thusis weakly lower semicontinuous.
The infimum of over is attained. A minimizing sequence is bounded in because is bounded below; weak compactness, Rellich strong convergence, and weak lower semicontinuity complete the direct-method argument.
Strong convergence on every bounded region combines with a tail estimate uniform in the sequence to give global strong convergence.
If local compactness makes every subsequential local limit agree with the global weak limit and the mass is uniformly tight, weak convergence upgrades to strong convergence.
Translations of one compactly supported bump have equal Sobolev norms. Widely separated translates have disjoint supports and remain a fixed positive distance apart in , obstructing compactness on an unbounded domain.
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.
For with , the dilationpreserves the norm and satisfies . Hence normalized functions on the line have Dirichlet-energy infimum zero, which no nonzero function attains.
A bounded function belongs to when
If with , then Fourier inversion, , and Cauchy-Schwarz giveThe weighted frequency integral converges precisely because .
For ,Apply the fundamental theorem of calculus to in each coordinate, multiply the resulting one-dimensional bounds, integrate, and use Cauchy--Schwarz.
C0(X) consists of continuous functions whose values become arbitrarily small outside compact sets.
With the supremum norm, 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.
Translations of a compactly supported bump can form an infinite family separated by a fixed sup-norm distance, disproving compactness.
The continuous dual is the normed space of bounded linear scalar-valued functionals on .
A linear functional on an ordered vector space of functions is positive when implies . Positivity implies monotonicity: gives .
The operator norm is .
The matrix 2-norm is the operator norm induced by the Euclidean norm:It equals the largest singular value of and is invariant under multiplication by orthogonal or unitary matrices.
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.
Coordinate pairing gives for , , and .
For and conjugate exponent , every bounded linear functional on has the formfor a unique , and its operator norm is .
On a finite measure space, a positive functional defines the finite measure . The Radon-Nikodym theorem gives with . The boundand - norm duality imply , after which density extends the integral representation to every .
Exponents are conjugate when , with the convention .
For nonzero , a normalized sequence proportional to attains equality in Hölder's inequality.
For suitable measurable and ,Pair the left side with a unit vector in the dual space, use Tonelli's theorem, and apply Hölder's inequality in .
The Banach space consists of scalar sequences tending to zero with the supremum norm.
The space consists of convergent scalar sequences with the supremum norm.
The map proves , equivalently .
A Banach space isomorphism is a bounded linear bijection with bounded inverse.
A spatial Fourier mode has the form and is an eigenfunction of every constant-coefficient spatial differential operator; in particular, .
The Schwartz space consists of smooth functions whose derivatives decay faster than every inverse polynomial. It is dense in every .
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.
A real distribution is positive when for every nonnegative test function . Every positive distribution has order zero and is represented locally by a positive measure.
For the convention , if , thenalmost everywhere. Reversing both exponential signs gives the equivalent opposite convention.
When , its inverse Fourier integral is continuous by dominated convergence. If is also continuous and Fourier inversion identifies the two functions almost everywhere, then they agree everywhere.
For the angular-frequency convention and sufficient decay,
For the angular-frequency convention,Consequently cosine transforms to the half-sum of the two shifted deltas, while sine transforms to their signed difference divided by .
For the angular-frequency convention,so multiplication by a phase factor in frequency translates a function in physical space.
The triangular functionhas angular-frequency Fourier transform .
The squared sinc function is nonnegative and integrable; it occurs as the Fourier transform of a triangular function.
The cutoffs increase pointwise to one, while their Fourier transforms are nonnegative and have integral under the angular-frequency convention.
If and , testing against triangular frequency cutoffs and using monotone convergence gives .
The convolution of integrable functions on the real line is
An approximate identity is a family of integrable kernels whose total mass is one and whose mass concentrates near zero as . Under standard hypotheses, converges to in norm and at almost every Lebesgue point.
For sequences , their discrete convolution is
For , Fubini gives .
For , induction in the convolution integral givesIts Fourier transform in the angular-frequency convention is .
If , then its Fourier transform is continuous and tends to zero as . Approximate in by a compactly supported smooth function; integration by parts makes the approximant's transform decay, while transfers the conclusion.
If a radial function on behaves like near zero and like at infinity, then its th power is locally integrable at zero exactly when and integrable at infinity exactly when . This follows from the radial measure factor .
For the unnormalized angular-frequency transform,After normalization, the Fourier transform extends unitarily to . For integrable and , inversion and Fubini give the identity directly.
Applying the Parseval identity to the repeated exponential convolution gives
For , the polylogarithm is ; contour formulas analytically continue it to a slit plane.
A complex-valued function is holomorphic on an open set when it has a complex derivative at every point of that set.
If a nonzero holomorphic function has expansionthen is the order or multiplicity of its zero at .
A simple zero has order one, equivalently and .
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.
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.
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.
An entire function is a holomorphic function whose domain is the whole complex plane.
Every nonconstant entire function takes every complex value with at most one exception.
Near an essential singularity, a holomorphic function takes every complex value, with at most one exception, infinitely often.
If is a holomorphic function on a connected domain andthen the Cauchy-Riemann equations force both components of to vanish. Hence is constant.
A domain is simply connected when every closed path in it can be continuously contracted to a point while remaining in the domain.
Every holomorphic function on a simply connected domain has a single-valued antiderivative. Fixing and settinggives a path-independent function with , because the Cauchy integral theorem makes the integral around every closed path zero.
If is the principal complex inverse sine, all values reached by analytic continuation areThis follows from the monodromy reflections about the branch values and .
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.
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.
For and real , integrate around a wide rectangle between the real axis and . The vertical contributions vanish as the width tends to infinity, so
The principal value uses symmetric truncation at infinity and symmetric deletion around real singularities.
For ,
for , and gives meromorphic continuation.
Integration by parts gives
The Euler--Mascheroni constant is
The reciprocal gamma function has the entire-product representation
The digamma function is the logarithmic derivativeLogarithmically differentiating the Weierstrass product gives
The digamma function is strictly increasing on the positive real axis, whileIt therefore has exactly one positive zero, and that zero lies in .
It follows by making the logarithm of the quotient entire and periodic; growth forces its periodic remainder to be constant.
For , Euler's beta function is
The sum-and-ratio change of variables in a product of gamma integrals gives
For , put and . Then and , separating radial gamma and ratio beta integrals.
The functional equation and give as away from the negative real axis.
The beta--gamma identity gives as through the right half-plane.
A map between Riemann surfaces is holomorphic when its expression in every pair of local complex coordinates is a holomorphic function.
A germ at is an equivalence class of holomorphic functions defined near , where two representatives are equivalent when they agree on some neighbourhood of .
The space of germs over a domain has basic open sheetsfor holomorphic on open . The projection restricts to a homeomorphism on each sheet, and these projections form holomorphic coordinate charts.
The evaluation map is holomorphic. On the sheet associated with , its coordinate expression is exactly the holomorphic function .
Over , the two-valued square root is represented byThe maps and identify this unbranched double cover analytically with the corresponding component of the space of germs.
If is a covering of a Riemann surface, compose every chart on an evenly covered open set with each local inverse sheet of . The resulting transition maps are those of , so they define a unique complex structure making a local biholomorphism.
Every simply connected Riemann surface is biholomorphic to the Riemann sphere, the complex plane, or the unit disc.
Every nonempty simply connected proper open subset of the complex plane is conformally equivalent to the unit disc.
The sphere has the Möbius group , the plane has the affine maps with , and the disc has the maps with .
The quotients of by free properly discontinuous conformal actions are , , and the complex tori . The translation group has respectively rank zero, one, or two.
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.
If embeds holomorphically in a compact Riemann surface , the embedding extends across zero and infinity to a degree-one holomorphic map from the Riemann sphere to . Hence is conformally the sphere.
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.
A real-valued function on a Riemann surface is harmonic when its expression in every holomorphic coordinate chart has vanishing planar Laplacian.
Under a holomorphic coordinate change ,so the condition is independent of the holomorphic chart.
A covering is regular when its deck group acts transitively on each fibre, equivalently when its fundamental-group subgroup is normal.
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.
For a nonconstant analytic map between compact connected Riemann surfaces, there is an integer such that for every ,
If a nonconstant holomorphic map has local-coordinate expressionthen is its local degree, multiplicity, or ramification index at .
For a nonconstant holomorphic map between compact connected Riemann surfaces, its degree isThe valency theorem says that this integer is independent of .
After cancelling common factors, the rational map on the Riemann sphere has degree . It is an analytic isomorphism exactly when this degree is one, equivalently when it is a Möbius transformation.
Let a nonconstant rational function have degree , let be its number of distinct finite poles, and let 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 at infinity into one of order . HenceIn particular, .
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.
If an elliptic function has degree and distinct poles in a fundamental parallelogram, then each pole order increases by one under differentiation, soBecause , this gives .
The rotation group of the octahedron has order . Its vertex, face-centre, and edge-centre stabilizers have orders , giving exceptional orbit sizes ; every other orbit has size .
Let a finite group of order act analytically on a compact Riemann surface, and let an invariant analytic map have degree . At a point with stabilizer of order , invariance forces the local multiplicity of to be at least . Its orbit has points, so it already contributes at least to the fibre multiplicity. The valency theorem forces equality and shows that every fibre is exactly one orbit.
For a degree- holomorphic map, .
The Riemann sphere is with two charts related by reciprocal coordinates.
Stereographic projection identifies a sphere minus one pole with the plane and supplies the standard charts of the Riemann sphere.
Let a finite group of conformal automorphisms of the Riemann sphere fix . The derivative representation at embeds into , so is cyclic. Averaging a local coordinate linearizes the action to , and the quotient has coordinate .
For a finite conformal group action, choose a disc around each point that meets only its stabilizer translates. The local cyclic quotient chart , where is the stabilizer order, gives the orbit space a Riemann-surface structure and makes the quotient map holomorphic.
For and , where , the rational functionis invariant under . Equality of two values factors as for and , so every fibre is exactly one dihedral orbit.
Removing a closed discrete set from a Riemann surface leaves an open complex one-manifold; if connected, it is again a Riemann surface.
A path in a surface can be perturbed inside coordinate discs to avoid finitely many prescribed points.
The punctured plane is a connected Riemann surface homeomorphic to a cylinder.
Polar coordinates give after taking logarithmic radius.
A homeomorphism to a complex manifold transports its atlas and thereby defines a complex structure on the source.
A nodal crossing locally consists of two complex branches meeting transversely and is not a one-dimensional complex manifold at the intersection.
If a punctured neighborhood has a different number of connected components from a punctured Euclidean ball, the point is not a manifold point.
A reducible complex curve is a union of proper complex subcurves; intersecting components can create singular points.
For a parameter , one form of the incomplete elliptic integral of the first kind isIts value depends on the choices of square-root branches and, under analytic continuation around the branch points, on the integration path.
For , the complete elliptic integral of the first kind is
For the complementary modulus ,
An elliptic function is a meromorphic function with two real-linearly independent periods . A fundamental cell is a half-open parallelogramOpposite 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.
If an elliptic function had no poles, it would be 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.
The Jacobi elliptic sine is the local inverse of the elliptic integral of the first kind: ifthen . Its analytic continuation is a meromorphic doubly periodic function.
For real , the period lattice of is generated byThe periods arise by composing the monodromy reflections of its inverse elliptic integral of the first kind.
For any finite value , apply the argument principle to around a fundamental parallelogram whose boundary avoids zeros and poles. Periodicity cancels the opposite-edge integrals, so the number of solutions of equals the fixed number of poles of , counting multiplicities.
The lattice sum is an even elliptic function with a double pole of principal part and zero residue at every lattice point .
The Weierstrass zeta function satisfies and is quasi-periodic: for every period , the difference is constant. It has a simple pole of residue one at each lattice point. Consequently, if , thenis an elliptic function.
Writing for the lattice Eisenstein sums, expansion about zero givesThus the constant coefficient vanishes and, for , the coefficient of is .
A holomorphic function near a compact set can be uniformly approximated by rational functions with poles in prescribed complementary components; connected complement permits polynomials.
If is compact with connected complement and is holomorphic near , then for each there is a polynomial such that .
A proper closed arc of a circle centered at zero has connected complement and stays away from zero. The polynomial Runge theorem therefore gives polynomials converging uniformly to on .
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.
For a closed piecewise smooth curve avoiding ,It is the net change of a continuous argument divided by and is an integer.
If is continuous and closed, choose a continuous lift such thatThen . This agrees with the contour-integral definition when the path is piecewise smooth.
A homotopy through closed paths avoiding the base point cannot change the integer winding number.
If closed continuous paths satisfy for every , then never vanishes. It is therefore a homotopy through closed paths in , and
For a polynomial of positive degree, on a sufficiently large circle its leading term dominates the remaining terms. The dominated-perturbation lemma therefore gives winding number to the loop . If had no zero, radial contraction of the input circle would map under to a homotopy with a constant loop in , which has winding number zero. This contradiction proves the Fundamental theorem of algebra.
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 from a loop of winding number one to a constant loop of winding number zero. Homotopy invariance of winding number rules this out.
A Bromwich contour is a vertical line in the complex frequency plane lying to the right of the singularities in the inverse Laplace integral.
When an inverse Laplace integrand contains 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.
A branch point is a point around which analytic continuation of a multivalued function returns a different value.
Letwhere has a simple zero at , and let be the limiting value of at on one sheet. Analytic continuation once around changes the sign of the square root and hence of . The continued primitive agrees at with the original one, so it is
If continuation around two square-root branch points acts on a primitive as and , thenThus pairs of branch-point loops generate additive periods of an inverse function.
Euler's formula states that .
The complex conjugate of is . It satisfies .
The complex unit circle is the subgroup under multiplication.
An isolated singularity at is a point at which a function is not holomorphic although it is holomorphic throughout some punctured neighbourhood of .
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.
An isolated singularity is removable when the function extends holomorphically across it, equivalently when its Laurent series has no negative-power terms.
A holomorphic function bounded on a punctured neighbourhood extends holomorphically across the puncture.
A holomorphic function tending to a finite limit at infinity becomes holomorphic at zero after reciprocal substitution.
If is holomorphic on andthen the singularity at zero is removable. The Laurent coefficient formula and Cauchy--Schwarz give for each , so every principal-part coefficient vanishes as .
A function has a pole of order at when extends holomorphically and is nonzero at .
A meromorphic function is holomorphic except at isolated poles. Equivalently, it is locally a quotient of two holomorphic functions whose denominator is not identically zero.
A rational function is a quotient of polynomials, with nonzero. After cancelling common factors, it defines a holomorphic map from the Riemann sphere to itself.
Over a field in which the denominator splits, a proper rational function is a sum of terms . The coefficient at a simple pole is obtained by multiplying by and evaluating at .
The order of vanishing of a nonzero rational function at a point is the exponent of a local parameter in its local factorization. A negative order is the order of a pole.
If has a pole at , then has an essential singularity there. The exponential series produces infinitely many negative Laurent powers from the nonzero principal part of .
An isolated singularity is essential when its Laurent series has infinitely many nonzero negative-power terms.
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 integrates complex functions along oriented curves and evaluates many real integrals through residues.
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 , prevents a global antiderivative.
Jordan's lemma controls exponential contour integrals on large semicircles and makes their arc contributions vanish.
Integrating along a straight segment bounds a complex integral by segment length times the supremum of the integrand.
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.
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.
When the complex power becomes an integer power, a collapsed Hankel contour extracts the coefficient of in the local Laurent series.
A univalent function is a holomorphic injective function.
Every bounded entire function is constant.
The image of every nonconstant entire function is dense in . If a disc about were omitted, then would be bounded and entire, so Liouville's theorem would make constant.
If an entire function satisfies away from the imaginary axis, then . Cauchy's formula on bounds every Taylor coefficient by a constant timeswhich tends to zero as .
An entire function bounded by a polynomial in the radius is itself a polynomial, by Cauchy estimates.
Cauchy’s derivative formula expresses derivatives as contour integrals with higher-order Cauchy kernels.
If is analytic and bounded by on , then Cauchy's derivative estimate giveson . Integrating along line segments gives a Lipschitz bound with the same constant in that smaller half-plane.
Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly.
A holomorphic function real on a boundary interval extends across it by conjugate reflection.
An upper-half-plane self-map is holomorphic and has positive imaginary part in the upper half-plane.
If a meromorphic function has no zeros or poles on a positively oriented boundary , thenwith zeros and poles counted by multiplicity.
If are holomorphic near the closure of a bounded domain and on its boundary, then and have the same number of interior zeros, counted with multiplicity. Apply the argument principle to the zero-free boundary homotopy .
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.
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.
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.
If is holomorphic near the closed unit disc, on its boundary, and at one interior point, then maps the open unit disc onto a set containing the open unit disc. Rouché's theorem first shows that has a zero and then that every with has one.
If is holomorphic and nonvanishing on a punctured disc, thenis the integer winding number of about zero. If it equals , the logarithmic derivative of has a removable singularity at zero.
A meromorphic function with finitely many specified simple zeros and poles is a rational product up to a nonzero holomorphic factor.
A nonconstant holomorphic function on a connected domain cannot attain a local maximum of its modulus at an interior point.
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.
Let be bounded and holomorphic on a half-plane and continuous on its closure. A bound on the boundary line propagates throughout the half-plane. Apply the bounded-domain principle to on expanding truncated half-discs, then let .
The value of a holomorphic function at the center of a closed disc equals its average around every concentric circle lying in the domain.
The Dirichlet beta function is and is the Dirichlet L-function for the nontrivial character modulo four.
Hankel residue extraction gives , , and values at other nonpositive integers from the Taylor coefficients of .
Parity in the functional equation forces the beta function's trivial zeros .
The functional equation may be written
Two holomorphic functions on a connected domain that agree on a set with an interior accumulation point agree everywhere.
If uniformly on large circles, then has residue at infinity in the large-contour sense and its counterclockwise large-circle integral tends to .
On a circle of radius , an integrand uniformly of order has contour integral of order by the estimation lemma.
A Fuchsian differential equation has only regular singular points, including possibly the point at infinity.
Substitution of a Frobenius behavior into the leading singular terms gives the indicial equation and its characteristic exponents.
When a second-order equation has a repeated indicial root and only one independent Frobenius series, a second solution has the local form
A singular point of a linear differential equation is irregular when it fails the regular-singular criterion. For , a pole of order greater than one in or greater than two in makes the point irregular.
A Papperitz or P-symbol lists the three regular singular points of a second-order equation and the two characteristic exponents at each.
For a second-order equation with three regular singular points, the sum of all six characteristic exponents is one.
A Möbius change of independent variable permutes the singular points while carrying their characteristic exponent pairs with them.
Multiplying a solution by adds to both exponents at the finite point and subtracts from both exponents at infinity, under the convention that an exponent at infinity corresponds to behavior .
The Gauss equation has singularities at and normalized exponent-zero solution at the origin.
With compatible branches and initially near ,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.
When is not an integer, a second local solution at zero is .
Away from resonant parameter cases, two independent local solutions at infinity areand the expression obtained by exchanging and . Every analytic continuation of a hypergeometric solution into their common domain is a constant linear combination of this basis.
The binomial series gives wherever the defining series converges, followed by analytic continuation.
Special parameter choices can identify a hypergeometric solution with an elementary solution of the same second-order differential equation by matching its local normalization.
The Laplace transform is and converts differentiation into multiplication by plus initial data.
Integration by parts giveswhenever the boundary term at infinity vanishes.
For , .
The delayed unit step has
The Heaviside function is zero before its switching time and one after it, allowing delayed signals to be written algebraically.
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, when is constant.
The centered Hardy-Littlewood maximal function on the real line is
The maximal averaging operator has weak type , controlling the measure of points where local averages are large.
Watson lemma obtains an endpoint Laplace-integral asymptotic expansion by integrating the local power expansion of its amplitude term by term.
Expanding at the endpoint and applying Watson's lemma gives
The upper incomplete gamma function is under the convention used here.
Repeated integration by parts generates successive inverse powers of a large endpoint, with a remainder expressed by the same integral with shifted parameters.
For fixed finite and , the omitted integral from zero to is negligible, so and Stirling's formula applies.
When with , the integral is dominated by its lower endpoint. Setting and applying Watson's lemma gives coefficients rational in .
Oscillatory integrals are led by points where the phase derivative vanishes, with a phase shift determined by the Hessian signature.
A stationary point of a smooth phase is nondegenerate when its Hessian matrix at is an invertible matrix. In one dimension, this condition is .
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.
For a large-parameter second-order ODE, WKB separates a rapidly varying exponential phase from a slowly varying transport amplitude.
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 on the allowed side.
For a one-dimensional energy and potential , a classical turning point satisfies . The classically allowed region has and supports an oscillatory leading-order WKB solution.
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 with the turning-point phase correction.
The semiclassical action integralis 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.
To classify infinity, set and classify . 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.
Writing in produces the Riccati equationExpanding in inverse powers of determines the exponential phase and algebraic prefactor recursively.
A Riccati equation is a first-order equation quadratic in the unknown function.
Numerical analysis studies algorithms for approximating mathematical problems and controlling their errors.
An arithmetic operation is one scalar addition, subtraction, multiplication, or division. Counting such operations gives a basic model of an algorithm's computational cost.
For a real monic quadratic , both roots lie in the closed unit disk whenwith the usual simplicity requirement for roots on the unit circle.
For bounded matrices,Commutativity removes this splitting error, though first-order substeps such as backward Euler retain their own quadratic local errors.
For , the composite midpoint rule with mesh width has local error and global error .
If with smooth near zero, the substitution givesThe transformed integrand is smooth, so an ordinary second-order midpoint rule recovers its usual convergence rate.
Given distinct nodes and values , there is a unique polynomial of degree at most taking those values.
The interpolating polynomial is
For distinct nodes,It is the leading coefficient of the interpolating polynomial through those nodes and obeys
Newton's nested interpolation form isA triangular divided-difference table computes all its coefficients using exactly divisions.
For samples , their discrete Fourier transform is
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 arithmetic operations.
The type-II discrete cosine transform of is
Reflect evenly to length by and . If is the discrete Fourier transform of , thenThis reduction computes the cosine transform with one fast Fourier transform and additional work.
One half-grid sine-transform convention is
If and is its type-II cosine transform, thenThus a sign modulation, one fast cosine transform, and output reversal give an sine-transform algorithm.
Every eigenvalue of a complex matrix lies in at least one disk
A symmetric conservative five-point diffusion matrix has row center and radius at most , where is the sum of its four nonnegative edge coefficients. If every coefficient is at most , its eigenvalues lie in . Forward Euler is therefore stable when .
A finite difference method replaces derivatives by algebraic combinations of values on a discrete space-time grid.
The centered unit-step approximation to a second derivative is
If , then is an integer linear combination of whenever is integer-valued. Successive differences recover the coefficients.
Von Neumann analysis inserts the Fourier mode into a constant-coefficient difference scheme. A one-step method is stable in the discrete -norm when its amplification factor satisfies for every resolvable wavenumber.
For a one-step translation-invariant recurrence, the amplification factor is the multiplier satisfyingThe Parseval identity converts the pointwise bound into non-growth of the discrete -norm.
For a schemethe Fourier ansatz gives the amplification polynomialIts roots are the amplification factors of that Fourier mode.
For the centered spatial second difference, the Crank-Nicolson diffusion scheme has amplification factorIt is stable for every .
The Courant number is the dimensionless ratio of physical propagation during one time step to one spatial grid spacing, commonly .
On interior points, the one-dimensional centered second-difference matrix with homogeneous Dirichlet boundaries has eigenvalues
If is the negative semidefinite Dirichlet discrete Laplacian, the Crank-Nicolson amplification matrixhas eigenvalues of modulus at most one for every .
If is the one-dimensional Dirichlet second-difference matrix on interior points, the two directional parts of the two-dimensional five-point Laplacian areThey commute, and is the Kronecker-sum discretization of the Laplacian.
The split stephas amplification matrixOn a common directional eigenvector its eigenvalue is .
On the by Dirichlet grid with , the exact discrete stability condition for isThe mesh-independent sufficient condition is , and the exact bound tends to as .
Adding the correctiongivesIts common-basis eigenvalues areSince , one has for every .
The Thomas algorithm is specialized Gaussian elimination for a tridiagonal linear system and uses linear time and storage in the system dimension.
An -node Gaussian quadrature rule uses orthogonal-polynomial roots and integrates every polynomial of degree at most exactly.
. These polynomials are orthogonal for weight and support spectrally accurate approximation.
The Chebyshev polynomial of the second kind is defined byIt is orthogonal on with weight and has roots for .
The number is an algebraic integer. If it is rational it must be an integer, which shows that is rational only for .
Among monic real polynomials of degree , uniquely minimizes the supremum norm on , attaining the value .
If has degree at most and on , alternation at the extrema of implies outside that interval.
The equation is obtained from the harmonic equation by .
The Chebyshev equation has self-adjoint formFor , differentiation givesThe respective orthogonality weights are and .
A Givens rotation is the identity except for a coordinate-plane blockChoosing maps the vector to .
Successive Givens rotations eliminate entries below a matrix diagonal. If their product is and is upper triangular, thenis a QR decomposition.
Gradient descent updates in the negative-gradient direction. For , this direction is the residual .
A differentiable function has a -Lipschitz gradient when
For with positive definite and residual , exact line search along usesThe objective error contracts by
For a positive-definite matrix , the conjugate gradient method chooses mutually -conjugate search directions. Starting with and , it usesand with
.
.
In exact arithmetic, conjugate gradients reaches the exact solution in at most the number of distinct eigenvalues of , and therefore in at most steps for an -dimensional system.
The heavy-ball iteration adds momentum to gradient descent:
For and ,Thus is a polynomial in applied to the initial residual and lies in the corresponding Krylov subspace.
For , , and , every error-propagation block has spectral radius at most , and equality occurs.
The Peano kernel theorem represents a linear approximation error that annihilates polynomials below degree as
At unit spacing, exactness through cubic polynomials gives
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 gives
For a two-periodic Fourier series sampled at equally spaced points, discrete averaging retains exactly the modes divisible by :If with , thenwhich is exponentially small.
For and Fourier truncation ,whereFor , 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.
If a real Fourier polynomial is strictly positive, then is Hermitian positive definite becausefor every nonzero .
If is Hermitian positive definite and is Hermitian, every eigenvalue of is real. Equivalently, is similar to the Hermitian matrix .
For with real nonzero , explicit Euler has amplification factor , whose modulus exceeds one for every .
The local truncation error of a one-step method is the defect after one numerical step started from the exact solution:A local error generally leads to global error under a uniform stability bound.
An implicit Runge-Kutta method defines one or more stages through equations involving those same stages.
The trapezoidal ODE rule averages vector fields at the old and new states and is A-stable.
The order is the highest power through which the one-step expansion matches the exact Taylor expansion.
A linear multistep method approximates an ODE using several previous solution and derivative values.
Zero-stability requires the roots of the first characteristic polynomial to lie in the closed unit disc, with unit-modulus roots simple.
The root condition is the polynomial criterion equivalent to zero-stability for a linear multistep method.
A consistent linear multistep method is convergent exactly when it is zero-stable.
Numerical linear algebra develops stable finite algorithms for matrix computations.
Gaussian elimination applies elementary row operations to reduce a linear system to triangular form. Dense elimination uses arithmetic operations.
An LDL decomposition of a real symmetric matrix iswhere is unit lower triangular and is diagonal. Symmetric elimination computes it one Schur complement at a time. The matrix is positive definite exactly when every diagonal pivot in is positive.
Machine precision is the characteristic relative rounding scale of a floating-point system.
For an approximate eigenpair , the residual is
If , the smallest operator-norm perturbation that makes an exact eigenpair has norm . One such rank-one perturbation is
Successive Householder reflections zero subdiagonal column entries and factor a matrix into an orthogonal factor and an upper-triangular factor.
Orthogonal Householder similarities reduce a real symmetric matrix to symmetric tridiagonal form.
The unshifted QR algorithm factors and forms , preserving eigenvalues while driving the matrix toward block diagonal form.
LetThenThus is a QR decomposition of the matrix power . In particular, the first columns of span the same subspace as the first columns of whenever those columns are independent.
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.
The accumulated orthogonal factor in QR iteration is the factor of a matrix power, so its leading columns span the same spaces as simultaneous power iteration.
Let a real symmetric matrix have an orthonormal eigenbasis , withWrite two starting vectors asFor two-column subspace iteration to recover
, the two projections onto that dominant subspace must be independent. The exact condition isRequiring each of the four coefficients to be nonzero does not imply this determinant condition.
, the two projections onto that dominant subspace must be independent. The exact condition isRequiring each of the four coefficients to be nonzero does not imply this determinant condition.
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.
A stationary iteration has the formwith a fixed iteration matrix . It converges for every initial vector exactly when the spectral radius satisfies .
Let be Hermitian positive definite. If is also Hermitian positive definite, then is invertible andThus the stationary iteration associated with the splitting converges from every initial vector.
If is symmetric positive definite and tridiagonal, set . Thenis positive definite. Applying the Householder-John theorem to and proves that the Jacobi iteration converges.
For a one-step method applied to , write . Its linear stability domain isForward Euler has , while backward Euler has and is A-stable.
A method is A-stable when its stability domain contains the closed left half-plane.
For the two-stage implicit Runge-Kutta method withthe stability function isFor ,The denominator has no zero in the closed left half-plane, so the method is A-stable for every real .
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 and , forward Euler requires , whereas backward Euler is stable for every positive step size.
Starting at the same value, forward and backward Euler have opposite leading local errors. Their half-difference therefore estimates either local error:For a first-order method, a local-tolerance controller consequently scales the next step by , usually with a safety factor.
Orthogonal polynomials of distinct degrees are orthogonal under a positive weighted inner product.
A monic orthogonal polynomial has leading coefficient one. For a fixed positive weight, there is a unique monic orthogonal polynomial of each degree.
A degree-n orthogonal polynomial for a positive interval weight has n simple zeros in the interval interior.
A Jacobi matrix is a symmetric tridiagonal matrix whose characteristic polynomials obey the orthogonal-polynomial three-term recurrence.
A quadrature rule approximates a weighted integral by a finite weighted sum of function values.
An -node quadrature rule already exact through degree is exact through degree exactly when its nodal polynomialis orthogonal to every polynomial of degree at most .
No -node quadrature rule for a positive interval weight can be exact through degree , because it evaluates the nonnegative nodal square as zero although its integral is positive.
If an -node quadrature rule for a positive interval weight is exact through degree , every weight is positive. Indeed, applying the rule to the square of the corresponding degree- Lagrange cardinal polynomial isolates that weight.
If each -node quadrature rule is exact through degree 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.
A quadrature rule is exact through degree precisely when its node weights reproduce the first moments of the integration measure.
Functions form an asymptotic sequence at when . The notationmeans that for every ,Successive division of the remainder by uniquely recovers every coefficient.
For a positive asymptotic sequence , define andThen term by term and is again an asymptotic sequence.
For and , the sequence is asymptotic, but for every .
Even if for every , an expansion in the scale need not induce one in . A perturbation of that is smaller than but larger than supplies a counterexample.
If a smooth real phase has a unique nondegenerate interior maximum at , thenFor a maximum at an endpoint with , the leading factor is .
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.
For a holomorphic phase , a saddle point is a critical point with . At a simple saddle, , and the quadratic Taylor term determines the local descent directions.
The modified Bessel equation of order zero isIts solution regular at zero and normalized by is the modified Bessel function .
Laplace's integral method seeks . Substitution into the differential equation and integration by parts transfer multiplication by into differentiation with respect to , producing a first-order equation for and endpoint conditions for .
For , substituting into and integrating by parts givesThe amplitude equation has solution . On both endpoint terms vanish, and hence
For the Laguerre equationthe ansatz givesThe contour must make the endpoint contributionvanish.
A finite contour based at an integrable branch point, or a closed contour around an integer-order pole, gives a solution analytic at . It is therefore a constant multiple of the unique Frobenius power-series solution there.
For , the amplitude has a pole at zero whose contour residue is a degree- Laguerre polynomial in . For , the amplitude has a pole at one whose residue is times a polynomial of degree .
The normalized order-zero function has the representation
Analytic continuation of an integral of a meromorphic function around a closed loop changes its value by times the winding-number-weighted sum of enclosed residues.
Two integration paths with the same endpoints differ by a closed-path period determined by the residues and winding numbers of their concatenation.
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.
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.
If is holomorphic on a neighbourhood of the closed disc and on its boundary, differentiating the Cauchy integral formula gives
On , choose the argument in . Thenis an analytic logarithm. Locally this follows by scaling the convergent series
For ,with value one at by continuity. A keyhole contour around the positive axis gives a jump factor , while the double pole at has residue .
The critical set of a differentiable map consists of the points where its derivative fails to have full rank.
A function has a local minimum at when throughout some neighbourhood of . For a twice differentiable function, its Hessian matrix at an interior local minimum is positive semidefinite.
A forced harmonic oscillator responds at both its natural and forcing frequencies.
A smooth function has continuous derivatives of every order.
A function can be differentiable even when its partial derivatives are discontinuous. For example,away from the origin, with value zero at the origin, is differentiable there while its partial derivatives oscillate along the coordinate axes.
If and its partial derivative are continuous for in a compact interval and in an open set, thenApply the fundamental theorem of calculus to the difference quotient in the th coordinate and use uniform continuity on a compact neighbourhood.
For , stationarity under fixed-endpoint variations gives
When a variational integrand has no explicit dependence on the independent variable, every extremal satisfies
A catenary is a curve of the form . It is the equilibrium shape of a uniform flexible chain in a uniform gravitational field.
When the values of are free at both fixed endpoint locations, integration by parts leaves the boundary term . Since the endpoint variations are arbitrary, stationarity requires
Forwith free terminal values, the Euler-Lagrange equations and natural conditions give a nonzero stationary family exactly when . It is
First-step analysis conditions on the first transition of a stochastic process, turning hitting probabilities, hitting times, and accumulated rewards into linear recurrence equations.
For a damped linear oscillator driven periodically, the steady-state response is the periodic particular solution left after homogeneous transients decay.
The Leibniz formula is the conditionally convergent alternating series
A trigonometric polynomial is a finite linear combination of complex exponentials , or equivalently of sines and cosines. If it vanishes on an interval, all its coefficients vanish.
For a function of period , its complex Fourier coefficients are
A Fourier cosine series has the formIt represents the even function obtained by reflecting its data across the origin.
For , the angular-frequency Fourier transform is
If a nontrivial uniform estimateholds for all integrable functions on , dilation by every forcesThus must be the conjugate exponent of .
As a tempered distribution on ,for a nonzero convention-dependent constant . One proof writesand applies the Fourier transform of a Gaussian before substituting .
With one common sign convention,Reversing the denominator reverses the sign.
With the standard Fourier convention, the integral of sinc squared over the real line equals pi.
Let solve with and . For Neumann boundary conditions, the Green function iswhere . The derivative jump is one. If , the Abel identity makes constant and symmetric.
A causal Green function vanishes before the source time and has the derivative jump required by a delta source.
Forwith the boundary conditions , , and , the higher-order Euler-Lagrange equation isIts solution splits as
For the clamped-free beam under uniform load and endpoint force, the additional minimized internal energy isIts derivative is the endpoint displacement caused by the endpoint force:
An analytic function on an annulus has a unique locally uniformly convergent expansionwhere is any positively oriented circle around in the annulus.
A linear differential equation is linear in the unknown function and its derivatives.
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.
For continuous , the initial-value problemhas a local continuously differentiable solution. If , use the constant solution; otherwise locally invert .
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.
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.
For , its prolongation to derivatives through order isIt generates a symmetry of exactly when vanishes on the equation manifold.
On jet coordinates,It differentiates a differential function along prolonged solution graphs.
The vector fields generateTheir second prolongations multiply the differential equation by , so they are infinitesimal Lie symmetries.
A boundary value problem asks for a differential-equation solution satisfying conditions at more than one point or boundary component.
A boundary condition prescribes values of an unknown function or its derivatives at the boundary of the domain of a differential equation.
A Dirichlet boundary condition prescribes the value of the unknown function on the boundary.
A Neumann boundary condition prescribes the outward normal derivative of the unknown function on the boundary.
A far-field boundary condition prescribes the limiting behavior of a solution as one or more spatial coordinates tend to infinity.
For , the initial-value problemhas the continuous piecewise differentiable solutionThe 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.
The iteration 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 . Consequently its uniform grid-point error is at most for arbitrarily long finite time intervals.
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.
A differential inequality bounds derivatives rather than specifying them exactly; comparison and integration turn it into bounds on the function.
A differentiable real function satisfyingon an interval has and , hence wherever the tangent remains finite.
A linear ordinary differential equation is linear in the unknown function and its derivatives.
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.
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.
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.
A second-order linear equation has the form .
If , then between consecutive zeros of a nontrivial solution ofthere is a zero of every nontrivial solution of , unless the two coefficients agree throughout that interval.
For solutions , the mixed WronskiansatisfiesIts endpoint signs between consecutive zeros of a positive force a zero of unless .
Kummer's equation isIts solution analytic at zero is the confluent hypergeometric functionWhen is not an integer, a second local solution is
For , the Wronskian satisfies .
Variation of parameters replaces the constants in a complementary solution by functions to construct a particular solution.
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.
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.
Forthe power ansatz gives the indicial equation . Distinct roots give ; a repeated root gives on a chosen logarithm branch.
The equation has polynomial solutions for nonnegative integer .
The polynomial solution of degree l normalized by P_l(1)=1 is the Legendre polynomial P_l.
For a contour enclosing ,
For fixed ,
For , the second derivative transforms as .
An integrating factor turns a first-order linear equation into an exact derivative.
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 occurs when forcing overlaps a homogeneous mode, requiring multiplication of the usual particular ansatz by an extra power or logarithm.
When forcing matches a natural frequency, the oscillator response acquires a linearly growing amplitude.
A linear differential system has vector form .
For a constant coefficient system, eigenvectors split the homogeneous equation into scalar exponential or power-law modes.
A state transition matrix maps the state of a homogeneous linear system between two times.
Encoding two planar components as one complex variable turns a rotation-dilation system into a scalar complex equation.
Integrating an equation through a Dirac impulse determines the jump in the derivative or state while nonsingular lower-order terms contribute no jump.
A first integral is a function of time, state, and derivatives that remains constant along every solution.
The residue of a meromorphic function at an isolated singularity is the coefficient of in its Laurent series. At a simple pole,
If is holomorphic on and inside a positively oriented closed contour except for finitely many isolated singularities inside it, then
Suppose a contour follows the real axis, closes in the upper half-plane, and avoids each simple real pole by a small semicircle above it. Each indentation is clockwise and tends toIf the large arc vanishes and there are no enclosed poles away from the real axis, the Cauchy principal value is therefore
Residues at symmetry-related poles can occur with opposite signs and cancel in the contour sum.
Pairing residues at opposite poles can produce an alternating series that converges in a larger half-plane than the corresponding absolutely summed residue series.
The spectral radius is the largest modulus of an operator's or matrix's eigenvalues:
A regular Sturm-Liouville eigenvalue problem has the formon a bounded interval with self-adjoint boundary conditions and positive weight .
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.
For a regular self-adjoint Sturm-Liouville problem, eigenfunctions are orthogonal in the weighted inner productand the coefficient of is .
A differential operator is self-adjoint when integration by parts makes its inner products symmetric on its domain.
A second-order scalar differential equation is in self-adjoint form when it can be writtenMultiplication by and integration by parts then gives an energy identity.
If and satisfy homogeneous separated boundary conditions, the Lagrange identity givesThus can be solved only if is orthogonal to the null mode .
The substitution turns time derivatives into and spatial diffusion into , reducing a reaction-diffusion PDE to an ODE system.
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.
For diffusion coefficient and leading-edge growth rate , a Fisher-KPP-type pulled front has minimum speed .
A pulled front is selected by growth and diffusion in its small-amplitude leading edge rather than by nonlinear dynamics behind the front.
Real analysis studies limits, continuity, differentiation, integration, and convergence for real-valued functions.
A function is Lipschitz continuous when there is a constant such thatfor all points in its domain.
A Lipschitz bound has the form . It controls the change of a function by the change of its argument.
The relation near a limit point means that there for some constant .
The real line is the set of real numbers equipped with its usual order, metric, and topology.
A real interval contains every real number lying between any two of its elements.
A bounded closed real interval has the form .
A real-valued continuous function on a nonempty compact space attains both its minimum and its maximum.
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.
A sequence is an ordered family indexed by the natural numbers; a series studies the partial sums of a sequence of terms.
A sequence is a function whose domain is usually the natural numbers.
A sequence in a metric space is bounded when all its terms lie in some ball of finite radius.
A binary sequence is a finite or infinite sequence whose terms belong to .
A real sequence is increasing when and decreasing when .
The limit superior of a real sequence isIt is the largest subsequential limit when the sequence is bounded.
A series is the formal or limiting sum of the terms of a sequence.
If a real sequence is subadditive, , and is bounded below in the required extended-real sense, then converges to .
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.
If are nonempty closed bounded intervals, then their intersection is nonempty. If their lengths tend to zero, the intersection consists of exactly one point.
If a bounded real sequence has the property that every convergent subsequence converges to the same number , then the full sequence converges to . Otherwise, a subsequence stays at least some fixed distance from ; Bolzano--Weierstrass gives it a convergent subsubsequence, contradicting the assumed uniqueness.
The Cesaro mean of the first terms of a sequence is their arithmetic average. If , then its Cesaro means also tend to .
For a positive sequence converging to , its cumulative geometric means converge to the same limit.
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.
The Fibonacci numbers satisfy , , and .
For nonnegative ,Cassini's identity is the adjacent-index special case.
For ,with the usual conventions for zero and infinity.
If for positive coefficients, then the coefficients have root limsup . Coefficients instead have root termsso their radius depends on the growth of relative to .
If the ordinary power series has radius , thenconverges for and diverges for . Behavior on the boundary depends on the coefficients.
The Cauchy product has coefficients . Absolute convergence permits regrouping and makes its sum the product of the two sums.
Inside its radius of convergence, a power series may be differentiated term by term, and the differentiated series has the same radius.
The first nonzero Taylor term controls a holomorphic function’s local magnitude and angular sign pattern.
A sequence of functions converges uniformly when one index makes close to the limit simultaneously for every .
A sequence of functions is uniformly Cauchy when, for every , all sufficiently late pairs satisfy
A sequence converges locally uniformly to when every point has a neighbourhood on which the convergence is uniform.
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.
For , a finite geometric series satisfies
A sequence is Cauchy when its terms become arbitrarily close to one another beyond some index.
Every Cauchy sequence of real numbers converges to a real number.
A series is conditionally convergent when it converges but does not converge absolutely.
If decreases to zero, then converges.
The alternating harmonic series 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.
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.
The series converges exactly when .
A series converges absolutely when the series of absolute values converges; absolute convergence implies convergence.
If and , then , so eventually and converges.
The Cauchy-Schwarz inequality gives
The total variation of a sequence is ; convergence alone does not make it finite.
Harmonic sums satisfy by integral comparison.
The notation means .
Ratios of exponential functions are controlled by comparing their linear growth exponents.
Stirling’s formula gives n factorial asymptotic to square root of 2 pi n times (n/e)^n.
A series converges when its sequence of partial sums converges to a finite limit.
For positive terms, if the ratio of two sequences tends to a finite positive number, their series either both converge or both diverge.
For nonzero terms, ifthen converges absolutely. If the ratio has a limit greater than one, the terms do not tend to zero and the series diverges.
For a positive sequence, if , thenFor this follows by taking logarithms and applying Cesàro averaging; the case follows from an eventual geometric upper bound.
A sequence converges uniformly when one index makes the approximation accurate at every point of the domain.
Calculus studies local change through derivatives and accumulated change through integrals.
An integral accumulates a function over a domain and may be defined as a limit of finite sums.
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.
If measurable functions satisfy almost everywhere, thenwhenever the two sides are defined.
Any two antiderivatives of the same function on a connected interval differ by an additive constant.
The Gaussian integral isSquaring the integral and changing to polar coordinates proves the formula.
A Gaussian function has the form with . Its integral and moments reduce by translation and scaling to the Gaussian integral.
The natural logarithm is the inverse of the real exponential function on the positive real numbers.
The hyperbolic cosine is .
The hyperbolic sine is .
The hyperbolic cotangent is where .
The hyperbolic secant is
The limit records the value approached by a function as its argument approaches a point.
A function is continuous at a point when its limit there equals its value.
A function is locally constant when every point has a neighbourhood on which the function is constant.
If is continuous and lies between and , then some satisfies .
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.
If a continuous satisfies , then the incrementssatisfy . Continuity forces to vanish somewhere on .
Every continuous injective function from a real interval to the real line is strictly monotone. A continuous bijection therefore has a continuous inverse.
If is continuous at and is continuous at , then is continuous at .
The continuity set of a function is the set of points where it is continuous. For real functions it is always a set.
A function is uniformly continuous when one input tolerance works at every point of its domain.
If two functions with the same limit bound a third nearby, the bounded function has that limit too.
The derivative is the limit of the difference quotient and gives the best linear approximation to local change.
Every derivative has the intermediate-value property, even when the derivative is discontinuous.
The higher-order product rule is
At zero, is not differentiable and is differentiable, while has derivative one.
For twice differentiable functions,
A function is increasing when implies .
A function is strictly increasing when implies . It is therefore injective and each value has at most one preimage.
Every real-valued monotone function on a real interval is differentiable with a finite derivative at almost every point.
Multivariable calculus extends differentiation and integration to functions of several variables.
A partial derivative differentiates a multivariable function with respect to one variable while holding the others fixed.
The gradient of a scalar-valued differentiable function is the vector of its first partial derivatives.
For a differentiable scalar function of several variables, the total differential is
The Hessian is the symmetric matrix of second partial derivatives; definiteness classifies nondegenerate critical points.
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 studies differentiation and integration of scalar and vector fields.
A vector field assigns a vector to every point of its domain.
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.
For in Cartesian coordinates,
The curl of has components
For a twice differentiable vector field ,
The Laplacian of a twice differentiable scalar field is the divergence of its gradient, ; it acts componentwise on vector fields.
For smooth vector fields ,andBoth follow by contracting two Levi-Civita symbols and applying the product rule.
In three dimensions, is zero when indices repeat and is the sign of the permutation otherwise.
Contracting one index gives
Contracting two Levi-Civita symbols gives
For linearly independent unit vectors with , a pairis attained by some unit vector exactly whenThe left-hand side is the squared norm of the least-norm vector having those two inner products.
On a simply connected open subset of , every continuously differentiable curl-free vector field is the gradient of a scalar potential.
A surface integral uses the area element induced by a parametrisation to integrate over a surface.
A parametrised surface has area element .
The graph has area element .
A flux integral measures flow through an oriented surface.
Green’s theorem converts circulation around a positively oriented planar boundary into the area integral of scalar curl.
A line integral integrates a scalar or tangential vector-field component along a curve.
Applying the divergence theorem componentwise to yields a boundary traction term and a volume contraction.
Tensor integration by parts transfers a derivative between tensor factors and introduces the boundary contraction with the outward normal.
The Jacobian matrix contains all first partial derivatives of a coordinate transformation.
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.
The change-of-variables formula multiplies an integral by the absolute determinant of the Jacobian.
Polar coordinates use , and area element .
A cardioid is an epicycloid with one cusp; a standard polar equation is .
Hyperspherical coordinates extend polar coordinates to dimensions with one radius and angles.
For , the spherical-coordinate curl follows from the orthogonal-coordinate scale factors .
The radius- ball in has volume .
The boundary of the radius- ball has area .
A map is differentiable at when it differs from an affine map with linear part by at .
A function is continuously differentiable when its derivative exists and varies continuously.
An invertible linear map is a linear bijection; its inverse is also linear.
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.
If a continuously differentiable map has invertible derivative at a point, it is a local continuously differentiable diffeomorphism there.
A local diffeomorphism restricts near every point to a diffeomorphism onto an open subset. In particular, every local diffeomorphism is an open map.
An open map sends every open set to an open set.
If and the partial derivative with respect to is invertible at , then the nearby zero set is the graph of a continuously differentiable function.
Implicit differentiation differentiates an identity and uses the chain rule to solve for derivatives of the implicitly defined function. In one dimension, when .
If an integrand is odd under a measure-preserving symmetry of the domain, its integral is zero.
Integration by parts is the integrated product rule: .
The hyperbolic tangent is sinh divided by cosh and approaches plus or minus one at the two infinities.
The identity implies
Riemann integration approximates area by upper and lower sums over finite partitions.
Every Riemann-integrable function on a compact interval can be approximated in by continuous functions. Consequently their integrals converge uniformly over all subintervals.
If Riemann-integrable functions satisfy throughout , thenThis follows directly by comparing their lower and upper Darboux sums, or their Riemann sums on a common sequence of refining partitions.
A bounded function is Riemann integrable exactly when for every some partition satisfies .
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.
A lower Darboux sum uses the infimum of a function on each partition interval, while an upper sum uses the supremum.
For a bounded function,The function is Riemann integrable exactly when these values agree.
For a nonnegative unbounded function, one truncation convention sets and asks that every be Riemann integrable and that have a finite limit as .
Under uniform length on , the position of the first occurrence of a fixed decimal digit hasThe set of expansions in which the digit never occurs has length zero, and
The supremum of a set of real numbers is its least upper bound.
A function is convex when its value on a line segment is at most the corresponding affine interpolation of endpoint values.
The pointwise maximum of finitely many convex functions is convex because
The softplus function is convex because
The absolute value function is convex by the triangle inequality:
A differentiable function is -strongly convex when
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.
The hessian matrix ofis positive semidefinite exactly whenThis region is convex: for it is the epigraph of in the positive quadrant, and for it is the closed first quadrant.
The perspective of f is g(t,x)=t f(x/t) for positive t and preserves convexity.
A vector is a subgradient when its supporting affine function lies below the convex function.
For a convex function and an integrable random variable ,The inequality is reversed when is concave.
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.
On the positive orthant,is strictly concave. Its unique maximizer on a compact convex feasible set satisfies the first-order inequalityequivalently
Measure theory supplies a rigorous language for size and integration on general spaces.
For a measurable set , its density at with respect to a locally finite measure iswhen this limit exists.
A point is a Lebesgue point of a locally Lebesgue integrable function whenThe Lebesgue differentiation theorem says that almost every point is a Lebesgue point.
For every Lebesgue measurable set , its Lebesgue density is at almost every point of and at almost every point of its complement.
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.
A property holds almost everywhere with respect to a measure when the set of points where it fails has measure zero.
A measurable function is Lebesgue integrable when .
For an integrable random variable and a sub-sigma-algebra , the conditional expectation is the almost-everywhere unique -measurable integrable random variable satisfyingfor every .
Taking expectation after conditioning recovers the original expectation. More generally, if , then
For nonnegative measurable functions,
On , the functionsconverge pointwise to zero but satisfy . Hence the two sides of Fatou lemma can be zero and one.
Ergodic theory studies the long-time statistical behaviour of measure-preserving transformations.
A measurable transformation preserves a measure when for every measurable set .
The invariant sigma-algebra of consists, modulo null sets, of measurable sets satisfying .
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.
Every orbit of an irrational rotation of the circle visits a half-open interval 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.
For a measure-preserving transformation and , the ergodic averagesconverge almost everywhere to , the conditional expectation on the invariant sigma-algebra. The limit has the same integral as .
On a finite measure space, the convergence in the Birkhoff ergodic theorem also holds in . Prove it first for bounded truncations by the dominated convergence theorem, then use the contraction and Fatou lemma to remove the truncation.
A measure on a sigma-algebra is a nonnegative, countably additive set function that assigns zero to the empty set.
For a measurable map , the pushforward of is the measureIt is the distribution induced on by mapping points distributed according to through .
If a measure has density with respect to , thenThis identity is the basis of likelihood ratios and importance sampling.
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.
For pairwise disjoint measurable sets , countable additivity means
A measure on is finite when .
The indicator function of a set is for and otherwise.
The indicator vector of a subset has coordinate on and coordinate outside .
A sigma-algebra on contains and is closed under complements and countable unions.
A pi-system is a nonempty family of subsets closed under finite intersections.
A Dynkin system contains , is closed under complements, and is closed under countable disjoint unions. Equivalently, it contains differences whenever are members.
Every Dynkin system containing a pi-system also contains the sigma-algebra . Equivalently, the Dynkin system generated by a pi-system equals the sigma-algebra it generates.
Fubini's theorem permits the order of integration of an absolutely integrable function on a product measure space to be exchanged.
If two measures agree on a generating -system and the space is a countable union of members having finite common measure, then they agree on the generated -algebra.
Lebesgue measure is the unique translation-invariant -finite Borel measure on normalized by . Translation invariance follows by comparing translated measure with on half-open intervals and applying uniqueness.
Every subset of a Lebesgue-null set is Lebesgue measurable and has measure zero.
The space consists of measurable functions with finite -norm, identified when they agree almost everywhere. This quotient is necessary because the integral seminorm vanishes on functions supported on null sets.
For , the Lebesgue space is complete in its usual norm.
The intersection is Banach for , but it is generally incomplete when equipped with the norm alone. Truncations of an function outside provide an -Cauchy sequence with no limit in the intersection.
Every nonnegative measurable function is the pointwise increasing limit of nonnegative simple functions, and its integral is the supremum of their integrals.
If nonnegative measurable functions satisfy almost everywhere, then , allowing the value infinity.
If almost everywhere and for one integrable function , then is integrable and .
Tonelli’s theorem permits interchange of integrals for a nonnegative measurable function, even before finiteness is known.
A measure is absolutely continuous with respect to when every -null set is also -null.
Let be continuous, strictly increasing, and satisfy Lusin condition N. Thendefines a measure on the Lebesgue measurable sets, and . Moreover,for .
If , then exactly when for every there is such that implies . For necessity, a contrary sequence with and has tail unions decreasing to a -null limsup; continuity from above for finite contradicts absolute continuity.
Two measures are mutually absolutely continuous when they have exactly the same null sets.
For sigma-finite measures, implies for a nonnegative measurable density unique almost everywhere.
Mutual absolute continuity is equivalent to the Radon--Nikodym derivative being finite and strictly positive almost everywhere.
A countable family of probability measures is dominated by any mixture with every and .
An increasing right-continuous function determines a measure byThe Radon-Nikodym theorem and Lebesgue differentiation theorem identify the density of the absolutely continuous part of this measure with almost everywhere.
On a finite measure space, for .
On , is integrable exactly when , providing sharp counterexamples between finite-measure spaces.
A partial differential equation relates a multivariable function to its partial derivatives.
The upper-half-space Poisson kernel gives the harmonic extension of boundary data.
The Poisson integral convolves boundary data with the Poisson kernel to produce a harmonic function in a half-space. In the upper half-plane,
A diffusion equation describes smoothing caused by a flux down spatial gradients.
Fick's first law says that diffusive flux points down the concentration gradient:Combining it with local conservation gives
.
.
The heat equationmodels diffusion with constant diffusivity .
The nonlinear equationis the potential form of the viscous Burgers equation. The substitution converts it to the heat equation .
For on the real line,has total mass one and converges to the delta distribution as decreases to zero. The solution is .
The centered normal densitiesform an approximate identity: for every , at almost every point where the Lebesgue differentiation theorem applies. Their Fourier transforms are in the angular-frequency convention.
For initial data on the real line, Fourier transformation or convolution with the fundamental solution gives
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,
The constant-coefficient one-dimensional advection-diffusion equation isTranslation to coordinates moving at speed reduces it to the heat equation.
The Cauchy problem with initial value has
A nonlinear diffusion equation lets diffusivity depend on the evolving field; the porous-medium equation is a standard example.
A reaction-diffusion system combines local reaction kinetics with spatial diffusion,
A morphogen reaction-diffusion equation models a spatial concentration by diffusion and local production or decay. Linearization at a homogeneous equilibrium replaces by .
If a fast inhibitor satisfies , then its spatial Fourier transform obeysSubstitution produces a nonlocal scalar equation for the activator.
For activator diffusion , local linear decay , and coupling , eliminating the fast inhibitor givesIts pattern-forming threshold is
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.
Let a stable two-species reaction equilibrium have Jacobianand diffusion matrix . A mode with has determinantSince its trace decreases with , diffusion-driven instability occurs exactly when
When , every spatial Fourier mode replaces the reaction Jacobian by . Each eigenvalue is shifted left by , so diffusion cannot destabilize a stable homogeneous equilibrium.
For reaction Jacobianthe critical diffusivity ratio isIt approaches one as the stable reaction system approaches marginality.
The Brusselator is a two-species autocatalytic reaction model. In the parametrizationits positive homogeneous equilibrium is .
For the spatially homogeneous Brusselator with parameters , set . The quadrilateralis a trapping region. The four inward-pointing tests use on , on , on , and on the sloping edge.
The spatially homogeneous Brusselator has its unique positive equilibrium at . Its Jacobian there has determinant and trace . If , the equilibrium is a repeller inside the Brusselator trapping region; the Poincare-Bendixson theorem then supplies a periodic orbit.
When and have diffusivities and , respectively, and the homogeneous Brusselator equilibrium is stable, a Turing instability occurs precisely whenAt onset,
A similarity solution combines independent variables into scale-invariant coordinates and reduces a PDE to an ODE.
Separation of variables seeks a product of one-variable factors, reducing a PDE to ordinary differential equations.
A polynomial ansatz determines an unknown polynomial solution by matching coefficients after applying the differential operator.
A harmonic function satisfies Laplace's equation .
A real-valued function on a simply connected domain is harmonic exactly whenfor some holomorphic function . One construction observes that is holomorphic and takes its holomorphic primitive.
A twice differentiable function is strictly subharmonic where . It cannot have an interior local maximum because the Hessian at such a maximum is negative semidefinite.
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 rules out an interior maximum immediately because the Hessian matrix there would be negative semidefinite.
A continuous function harmonic on a bounded domain attains its maximum and minimum on the boundary. Apply the strictly subharmonic result to and let , then apply the same argument to .
A bounded harmonic function on the plane is constant. More generally, an entire harmonic function bounded on either side is constant: write it as by harmonic function as the real part of a holomorphic function, then apply the Liouville theorem to or .
Every nonconstant harmonic function is unbounded above and below by the Harmonic Liouville theorem. The intermediate value theorem then gives .
Let be the Weierstrass elliptic function forBecause is a period, the formulais independent of the branch of the complex logarithm. It is a nonconstant harmonic function onand its other period gives .
A harmonic polynomial is a polynomial annihilated by the Laplacian.
A harmonic conjugate v makes u+iv holomorphic; it exists globally on simply connected domains.
On a disc, the closed one-form has a potential when . The Cauchy--Riemann equations then make holomorphic.
The function is 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 , whose integral around the unit circle is , whereas the integral of a derivative around a closed curve is zero.
Every harmonic function is locally the real part of a holomorphic function and is therefore infinitely differentiable.
Harmonic functions admit locally convergent power-series expansions.
A harmonic function on a connected domain that vanishes on a nonempty open subset vanishes everywhere.
A smooth bump function is a smooth function with compact support. The flat exponential construction gives bumps supported in prescribed balls.
Two nonzero functions with disjoint supports have identically zero pointwise product. Separated continuous tent functions or smooth bumps provide examples.
The Laplace operator is the divergence of the gradient, .
Laplace's equation isIts solutions are harmonic functions.
In plane polar coordinates, Laplace's equation isSeparation of variables gives radial powers for positive angular modes and for the zero mode.
For radial in dimensions, .
The spherical-coordinate Laplacian splits into a radial operator and the sphere’s angular Laplacian divided by .
Axisymmetric harmonic functions separate into radial powers times Legendre polynomials.
A Laplacian eigenfunction satisfies together with specified boundary conditions.
A Dirichlet Laplacian eigenfunction vanishes on the boundary and satisfies .
The wave equation models finite-speed propagation and second-order oscillation.
Small displacements in an isotropic elastic solid satisfy a vector wave equation governed by density and the two Lamé moduli.
Divergence of the elastic equation isolates dilatational motion, while curl isolates rotational motion, producing P and S wave equations with different speeds.
A P-wave is longitudinal, carries nonzero dilatation, and has speed in an isotropic solid.
In a longitudinal plane wave, displacement is parallel to the wavevector and the curl vanishes.
An S-wave is transverse, carries rotation without dilatation, and has speed .
In a transverse plane wave, displacement is perpendicular to the wavevector and its divergence vanishes.
An SV-wave is polarized in the vertical plane containing its wavevector and perpendicular to that wavevector.
An SH-wave is polarized horizontally and perpendicular to the vertical propagation plane.
For a layer with rigid lower boundary and traction-free upper boundary, an SH mode
hasand dispersionIts cutoff is , its phase speed is , and its group speed is .
hasand dispersionIts cutoff is , its phase speed is , and its group speed is .
At a perfectly bonded interface, all displacement and traction components are continuous.
An obliquely incident in-plane P-wave generally produces reflected and transmitted P and SV waves; SH polarization decouples.
Phase matching fixes a common frequency and tangential wavenumber for every incident, reflected, and transmitted elastic mode.
For two inviscid media, continuity of normal displacement and normal stress determines the reflected and transmitted longitudinal amplitudes.
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.
A stretched uniform string obeys y_tt=c^2 y_xx with wave speed equal to the square root of tension over density.
With fixed endpoints, separates into sine modes whose amplitudes satisfy
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.
A normal mode oscillates at one frequency with a fixed spatial eigenfunction.
Orthogonality separates total quadratic wave energy into a sum of independent modal energies.
A node is a point where a standing-wave eigenfunction vanishes at all times.
Fourier transformation in space turns into the independent oscillators .
The Cauchy problem has
If , , and the odd function is supported on , thenThe interval then contains the entire support, whose integral vanishes by oddness.
The wavenumber is the spatial angular frequency of a monochromatic wave. A factor has wavelength .
A dispersion relation gives angular frequency as a function of wavenumber for monochromatic waves admitted by a linear wave equation.
For a normal mode proportional to , the real part is its exponential growth rate. A positive growth rate signals linear instability.
For a dispersion relation , the phase velocity is and the group velocity is . For the Klein-Gordon relation at ,
A wave crest is a point of constant phase. For a monochromatic wave it propagates at the phase velocity.
A wave packet is a localized superposition of nearby wavenumbers. Its envelope propagates at the group velocity to leading order.
The one-dimensional Klein-Gordon equationhas positive-frequency dispersion relation .
For zero initial velocity and Fourier amplitude , observation along with selects the two stationary wavenumbers , whereWriting , the leading oscillation has angular frequency and amplitude proportional to .
Green’s second identity is the divergence theorem applied to phi grad psi minus psi grad phi.
Green's third identity represents a function inside a domain through its boundary values, normal derivatives, and a volume term involving its Laplacian.
A Lie point symmetry is a continuous transformation of independent and dependent variables that maps solutions of a differential equation to solutions.
A one-parameter point transformation has infinitesimal generatorIts flow recovers the finite transformations.
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 when its required prolongation sends to zero on the solution manifold.
A scaling is a symmetry when every term of the equation transforms with the same weight.
The flow of is . Its second prolongation scales both and with weight minus two, so it preserves .
An invariant of a one-parameter symmetry group becomes a similarity variable that reduces a PDE to an ODE.
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 .
The vector fieldgenerates the local transformationsIts second prolongation sends to times that expression, so it is a Lie point symmetry of the potential Burgers equation.
A similarity variable is constant along symmetry-group orbits and parametrizes group-invariant solutions.
Distribution theory extends functions to continuous linear functionals on test functions, allowing generalized derivatives and point sources.
A distributional identity is an equality that holds after both sides act on every smooth compactly supported test function.
The Dirac delta is the distribution defined by .
For nonzero , in the distributional sense.
Calculus of variations finds stationary points of functionals, often among functions or regular curves.
A functional is a function whose inputs are themselves functions, curves, or other elements of a function space.
An energy functional assigns an integral energy to a function. For the modified Helmholtz equation, a natural example is
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.
A variation of a function is a one-parameter family , where satisfies the required boundary conditions.
If a continuous function satisfiesfor every smooth compactly supported test function , then on .
A point is stationary for a differentiable function or functional when its first derivative or first variation vanishes in every admissible direction.
Among functions with fixed boundary values, the solution of with uniquely minimizes the weighted energy
Fermat-type optical paths in cylindrical coordinates extremize refractive index times Euclidean arclength.
A helical extremal has constant radius and a polar angle linear in axial position.
Noether’s theorem associates a conserved quantity to each continuous variational symmetry.
The second variation is the quadratic term in a functional's expansion along ; positivity on admissible variations is a local-minimum test.
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.
A conjugate point marks a nonzero endpoint-vanishing Jacobi variation and the loss of positive definiteness of the second variation.
If a continuously differentiable -periodic function has mean zero, thenEquality holds exactly for .
A fixed-point theorem gives conditions under which a map has a point satisfying .
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.
A topological space has the fixed-point property when every continuous self-map has a fixed point.
If is a homeomorphism and has the fixed-point property, then a continuous induces . A fixed point of the conjugate map is carried by to a fixed point of .
For continuous , the function is nonnegative at zero and nonpositive at one, so the intermediate value theorem gives a fixed point.
A contraction of a nonempty complete metric space has one fixed point, and every orbit converges to it geometrically.
Suppose are contractions of one complete metric space with a common constant , and depends continuously on for each fixed . Their fixed points depend continuously on , because for a fixed parameter ,
If some iterate is a contraction on a complete metric space, then itself has exactly one fixed point: uniqueness for forces to fix its fixed point.
For the Newton map near a simple root ,Bounds and make on a sufficiently small closed interval around . The contraction mapping theorem then gives the unique local fixed point.
Uniform approximation minimizes the supremum norm of the difference between a target function and an approximant.
A degree-at-most- polynomial is a best uniform approximation precisely when its error attains alternating extrema of equal magnitude at at least ordered points.