Codex Wiki
OurBigBook.com
Site
Source code
Analysis
Home
Mathematics
Area of mathematics
OurBigBook.com
Words: 23k
Articles: 1k
Table of contents
23k
1k
Topological analysis
Analysis
2k
45
Brouwer fixed-point theorem
Topological analysis
90
3
Polynomial root from a disk self-map
Brouwer fixed-point theorem
27
Poincare-Miranda theorem
Brouwer fixed-point theorem
52
1
Clamped-map proof of planar path crossing
Poincare-Miranda theorem
27
No-retraction theorem
Topological analysis
21
Metric space
Topological analysis
1k
35
Convergence in a metric space
Metric space
24
Sequential characterization of continuity in metric spaces
Metric space
46
Sequential compactness of a compact metric space
Metric space
21
Normality of every metric space
Metric space
39
Bounded-continuous-function characterization of compact metric spaces
Metric space
46
Completeness
Metric space
14
Triangle inequality
Metric space
14
Euclidean distance
Metric space
8
Diameter of a metric space
Metric space
13
Uniform continuity
Metric space
67
2
Heine-Cantor theorem
Uniform continuity
15
Uniform limit theorem for uniformly continuous functions
Uniform continuity
33
Equivalent metrics
Metric space
96
2
Equivalent metrics need not be bi-Lipschitz equivalent
Equivalent metrics
31
Equivalent codomain metrics need not preserve uniform convergence
Equivalent metrics
41
Distance to a set
Metric space
17
Isolated point
Metric space
29
Complete metric space
Metric space
873
14
Cantor intersection theorem
Complete metric space
33
Baire category theorem
Complete metric space
608
7
Space of smooth functions on a compact interval
Complete metric space
187
3
Closed-subspace completeness theorem
Complete metric space
29
Hausdorff distance
Metric space
101
2
Closed singleton embedding in a Hausdorff hyperspace
Hausdorff distance
62
1
Equicontinuous family of functions
Topological analysis
130
3
Arzela-Ascoli theorem
Equicontinuous family of functions
92
2
Diagonal subsequence for locally uniform convergence
Arzela-Ascoli theorem
62
1
Functional analysis
Analysis
3k
107
Integral operator
Functional analysis
31
Normed vector space
Functional analysis
249
10
L1 norm
Normed vector space
10
Euclidean norm
Normed vector space
32
1
Euclidean ball
Euclidean norm
23
Supremum norm
Normed vector space
28
Surjective isometry of normed vector spaces
Normed vector space
120
3
Mazur-Ulam theorem
Surjective isometry of normed vector spaces
78
1
Nonsurjective isometry need not preserve midpoints
Surjective isometry of normed vector spaces
29
Banach space
Normed vector space
40
1
Uniform boundedness principle
Banach space
25
Stone-Weierstrass theorem
Functional analysis
85
2
Weierstrass approximation theorem
Stone-Weierstrass theorem
45
1
Bernstein polynomial
Weierstrass approximation theorem
33
Open mapping theorem (functional analysis)
Functional analysis
48
1
Bounded inverse theorem
Open mapping theorem
21
Diagonal operator on sequence space
Functional analysis
58
1
Unbounded inverse on a nonclosed operator range
Diagonal operator on sequence space
23
Graph of a linear operator
Functional analysis
173
5
Closed graph theorem
Graph of a linear operator
158
4
Closed graph theorem from the bounded inverse theorem
Closed graph theorem
26
Closed-graph proof for a factored operator
Closed graph theorem
25
Incomplete-domain counterexample to closed-graph factorization
Closed graph theorem
24
Continuity of a positive linear map into a dual space
Closed graph theorem
67
Closed-range criterion for an injection
Functional analysis
25
Neumann-series perturbation
Functional analysis
22
Density by annihilators
Functional analysis
19
Hilbert space
Functional analysis
556
21
Closest point theorem in a Hilbert space
Hilbert space
77
1
Orthogonal decomposition by a closed subspace
Closest point theorem in a Hilbert space
35
Riesz representation theorem
Hilbert space
162
5
Adjoint operator
Riesz representation theorem
144
4
Weak convergence in a Hilbert space
Hilbert space
237
8
Coordinate criterion for weak convergence in a separable Hilbert space
Weak convergence in a Hilbert space
26
Weak subsequence of a bounded Hilbert-space sequence
Weak convergence in a Hilbert space
32
Weak lower semicontinuity of the Hilbert norm
Weak convergence in a Hilbert space
29
Radon-Riesz theorem
Weak convergence in a Hilbert space
58
1
Weakly null orthonormal sequence
Weak convergence in a Hilbert space
23
Mazur lemma
Weak convergence in a Hilbert space
25
Norm-closed convex set is weakly closed
Weak convergence in a Hilbert space
26
Orthonormal sequence
Hilbert space
72
3
Hilbertian basis
Orthonormal sequence
46
1
Bessel inequality
Orthonormal sequence
12
Compact operator
Functional analysis
137
5
Hilbert-Schmidt operator
Compact operator
23
Finite-rank operator
Compact operator
9
Spectral theorem for compact Hermitian operators
Compact operator
51
1
Finite-rank truncation of a compact Hermitian operator
Spectral theorem for compact Hermitian operators
20
Coordinate-projection approximation of a compact operator
Compact operator
38
Sublinear functional
Functional analysis
15
Hahn-Banach theorem
Functional analysis
147
3
Canonical embedding into the bidual
Hahn-Banach theorem
40
Singular functional on L infinity
Hahn-Banach theorem
58
Hahn-Banach separation theorem
Hahn-Banach theorem
27
Banach-Alaoglu theorem
Functional analysis
15
Lax-Milgram theorem
Functional analysis
92
2
Weak Dirichlet problem for the massive Laplacian
Lax-Milgram theorem
57
1
Compact massive-Laplacian resolvent
Weak Dirichlet problem for the massive Laplacian
27
Sobolev space
Functional analysis
555
21
Sobolev derivative estimate
Sobolev space
19
Massive Laplacian isomorphism on Sobolev spaces
Sobolev space
20
Compactness from bounded support and an H2 bound
Sobolev space
34
Sobolev trace theorem
Sobolev space
70
1
Failure of an Lp hyperplane trace
Sobolev trace theorem
32
Regularity gain for one plus an even power of the Laplacian
Sobolev space
23
Sobolev embedding theorem
Sobolev space
338
13
Rellich-Kondrashov compactness theorem for H01
Sobolev embedding theorem
196
7
Noncompactness of a Sobolev embedding by translation
Sobolev embedding theorem
92
2
Hölder space
Sobolev embedding theorem
38
1
Ladyzhenskaya inequality in two dimensions
Sobolev space
30
Space of continuous functions vanishing at infinity
Functional analysis
83
2
Completeness of continuous functions vanishing at infinity
Space of continuous functions vanishing at infinity
49
Uniform square-root perturbation under linear growth
Space of continuous functions vanishing at infinity
21
Noncompactness by separated translates
Functional analysis
19
Continuous dual space
Functional analysis
195
7
Positive linear functional
Continuous dual space
26
Operator norm
Continuous dual space
35
1
Matrix 2-norm
Operator norm
26
Completeness of the dual space
Continuous dual space
28
Duality of sequence spaces
Continuous dual space
12
Duality of Lp spaces
Continuous dual space
78
1
Positive functional representation on Lp
Duality of Lp spaces
47
Holder inequality
Functional analysis
76
3
Conjugate exponents
Holder inequality
12
Norming vector for Holder inequality
Holder inequality
15
Minkowski integral inequality
Holder inequality
34
Space of sequences converging to zero
Functional analysis
42
2
Convergent sequence space
Space of sequences converging to zero
27
1
Absorption of a finite-dimensional summand by c0
Convergent sequence space
15
Banach space isomorphism
Functional analysis
12
Fourier analysis
Analysis
581
26
Fourier mode
Fourier analysis
22
Schwartz space
Fourier analysis
80
2
Tempered distribution
Schwartz space
54
1
Positive distribution
Tempered distribution
29
Fourier inversion theorem
Fourier analysis
121
4
Continuous version of Fourier inversion
Fourier inversion theorem
32
Fourier transform of a derivative
Fourier inversion theorem
9
Inverse Fourier transforms of phase factors
Fourier inversion theorem
31
Translation property of the Fourier transform
Fourier inversion theorem
21
Fourier transform of a triangular function
Fourier analysis
82
4
Triangular function
Fourier transform of a triangular function
10
Squared sinc function
Fourier transform of a triangular function
18
Triangular frequency cutoff
Fourier transform of a triangular function
43
1
Positive-Fourier-transform L1 bound
Triangular frequency cutoff
22
Convolution
Fourier analysis
94
4
Approximate identity
Convolution
39
Discrete convolution
Convolution
10
Convolution theorem
Convolution
34
1
Gamma density from repeated exponential convolution
Convolution theorem
22
Riemann-Lebesgue lemma
Fourier analysis
108
2
Radial power integrability criterion
Riemann-Lebesgue lemma
47
Interpolation between L2 and Linfinity by a pointwise bound
Riemann-Lebesgue lemma
18
Plancherel theorem
Fourier analysis
34
Parseval identity
Fourier analysis
30
1
Even rational Parseval integral
Parseval identity
13
Fourier transform of a Gaussian
Fourier analysis
10
Complex analysis
Analysis
4k
182
Polylogarithm
Complex analysis
17
Holomorphic function
Complex analysis
237
10
Order of a zero of a holomorphic function
Holomorphic function
35
1
Simple zero
Order of a zero of a holomorphic function
11
Analytic continuation
Holomorphic function
119
3
Monodromy theorem
Analytic continuation
25
Analytic continuation by contour deformation
Analytic continuation
43
Multivalued inverse hyperbolic sine
Analytic continuation
21
Entire function
Holomorphic function
42
2
Little Picard theorem
Entire function
31
1
Analytic function with image in an affine real line
Holomorphic function
24
Simply connected domain
Complex analysis
54
1
Primitive of a holomorphic function on a simply connected domain
Simply connected domain
31
Complex inverse sine
Complex analysis
50
1
Branches of the complex inverse sine
Complex inverse sine
31
Morera's theorem
Complex analysis
25
Cauchy integral theorem
Complex analysis
72
1
Gaussian contour translation
Cauchy integral theorem
34
Cauchy principal value
Complex analysis
20
1
Principal-value beta integral
Cauchy principal value
6
Principal-value residue rule
Complex analysis
12
Gamma function
Complex analysis
214
15
Gamma function recurrence
Gamma function
5
Bose integral
Gamma function
14
Euler--Mascheroni constant
Gamma function
6
Weierstrass product for the reciprocal gamma function
Gamma function
13
Digamma function
Gamma function
56
2
Trigamma function
Digamma function
11
Positive zero of the digamma function
Digamma function
29
Gamma reflection formula
Gamma function
2
Gamma duplication formula
Gamma function
21
Beta function
Gamma function
60
3
Beta--gamma identity
Beta function
33
1
Logarithmic moments of the Cauchy kernel
Beta function
19
Gamma function asymptotic at zero
Gamma function
26
1
Diagonal beta-function asymptotic at zero
Gamma function asymptotic at zero
11
Riemann surfaces
Complex analysis
1k
40
Holomorphic map
Riemann surfaces
20
Germ of a holomorphic function
Riemann surfaces
131
3
Space of germs of holomorphic functions
Germ of a holomorphic function
100
2
Complex structure lifted through a covering map
Riemann surfaces
48
Uniformization theorem
Riemann surfaces
167
5
Riemann mapping theorem
Uniformization theorem
15
Automorphisms of simply connected Riemann surfaces
Uniformization theorem
31
Riemann surfaces uniformized by the complex plane
Uniformization theorem
35
Plane domain with two omitted points is hyperbolic
Uniformization theorem
33
Compact Riemann surface containing an embedded punctured plane
Uniformization theorem
35
Identity theorem on a Riemann surface
Riemann surfaces
32
Harmonic function on a Riemann surface
Riemann surfaces
42
1
Conformal invariance of harmonicity
Harmonic function on a Riemann surface
21
Regular covering map
Riemann surfaces
20
Complete analytic function
Riemann surfaces
26
Valency theorem
Riemann surfaces
376
8
Local degree of a holomorphic map
Valency theorem
24
Degree of a holomorphic map
Valency theorem
29
Degree of a rational map of the Riemann sphere
Valency theorem
108
1
Degree of an elliptic function
Valency theorem
76
1
Octahedral rotation orbits on the Riemann sphere
Valency theorem
33
Degree-sized invariant separates finite-group orbits
Valency theorem
78
Riemann-Hurwitz formula
Riemann surfaces
8
Riemann sphere
Riemann surfaces
164
4
Stereographic projection
Riemann sphere
20
Local cyclic quotient of a Riemann surface
Riemann sphere
93
1
Orbit-separating invariant for the standard dihedral action on the Riemann sphere
Riemann sphere
38
Punctured Riemann surface
Riemann surfaces
64
3
Path avoidance in a surface
Punctured Riemann surface
17
Punctured complex plane
Punctured Riemann surface
25
1
Transport of a complex structure
Riemann surfaces
18
Nodal crossing
Riemann surfaces
61
2
Topological manifold local obstruction
Nodal crossing
23
Reducible complex curve
Nodal crossing
17
Elliptic integral of the first kind
Complex analysis
57
2
Complete elliptic integral of the first kind
Elliptic integral of the first kind
21
1
Complementary complete elliptic integral of the first kind
Complete elliptic integral of the first kind
8
Elliptic function
Complex analysis
315
9
Period lattice
Elliptic function
29
1
Fundamental parallelogram of a period lattice
Period lattice
18
Nonconstant elliptic function has a pole
Elliptic function
42
Jacobi elliptic sine
Elliptic function
50
1
Period lattice of the Jacobi elliptic sine
Jacobi elliptic sine
25
Value multiplicity of an elliptic function
Elliptic function
45
Weierstrass elliptic function
Elliptic function
96
2
Weierstrass zeta function
Weierstrass elliptic function
41
Laurent coefficients of the Weierstrass elliptic function
Weierstrass elliptic function
29
Runge theorem
Complex analysis
142
3
Polynomial Runge theorem
Runge theorem
118
2
Polynomial approximation of the reciprocal on a proper circular arc
Polynomial Runge theorem
33
Pointwise approximation by a Runge exhaustion
Polynomial Runge theorem
57
Winding number
Complex analysis
61
1
Winding number of a continuous closed path
Winding number
32
Homotopy invariance of winding number
Complex analysis
162
3
Dominated perturbation preserves winding number
Homotopy invariance of winding number
29
Winding-number proof of the fundamental theorem of algebra
Homotopy invariance of winding number
69
Winding-number proof of the no-retraction theorem
Homotopy invariance of winding number
49
Bromwich contour
Complex analysis
76
1
Bromwich inversion with a square-root branch cut
Bromwich contour
52
Branch point
Complex analysis
104
2
Monodromy reflection at a square-root branch point
Branch point
86
1
Translation generated by two square-root monodromy reflections
Monodromy reflection at a square-root branch point
31
Complex number
Complex analysis
87
7
Real part
Complex number
10
Imaginary part
Complex number
10
Euler's formula
Complex number
6
Modulus
Complex number
8
Complex conjugate
Complex number
14
Complex plane
Complex number
25
1
Complex unit circle
Complex plane
11
Isolated singularity
Complex analysis
353
13
Classification of isolated singularities
Isolated singularity
295
11
Removable singularity
Classification of isolated singularities
86
3
Pole
Classification of isolated singularities
169
5
Essential singularity
Classification of isolated singularities
15
Non-isolated singularity
Isolated singularity
32
Contour integration
Complex analysis
153
6
Period obstruction to a holomorphic antiderivative
Contour integration
32
Jordan lemma
Contour integration
15
Complex line integral estimate
Contour integration
18
Hankel contour
Contour integration
73
2
Hankel analytic continuation
Hankel contour
24
Residue extraction by a Hankel contour
Hankel contour
22
Univalent function
Complex analysis
8
Liouville theorem
Complex analysis
91
3
Dense image of a nonconstant entire function
Liouville theorem
34
Entire function under a horizontal inverse-square-root bound
Liouville theorem
34
Polynomial growth theorem for entire functions
Liouville theorem
17
Cauchy derivative formula
Complex analysis
49
1
Lipschitz bound inside a bounded analytic half-plane
Cauchy derivative formula
37
Locally uniform convergence of holomorphic functions
Complex analysis
14
Schwarz reflection principle
Complex analysis
14
Upper half-plane self-map
Complex analysis
14
Argument principle
Complex analysis
252
6
Rouché's theorem
Argument principle
187
4
Hurwitz's theorem
Rouché's theorem
28
Open mapping theorem (complex analysis)
Rouché's theorem
66
1
Unit-disc image from a boundary modulus lower bound
Rouché's theorem
52
Integer residue of a logarithmic derivative
Argument principle
39
Meromorphic function with prescribed zeros and poles
Complex analysis
21
Maximum modulus principle
Complex analysis
97
2
Maximum modulus principle on a bounded domain
Maximum modulus principle
77
1
Bounded half-plane maximum principle
Maximum modulus principle on a bounded domain
40
Mean value property for holomorphic functions
Complex analysis
24
Dirichlet beta function
Complex analysis
60
3
Special values of the Dirichlet beta function
Dirichlet beta function
22
Trivial zero of a Dirichlet L-function
Dirichlet beta function
13
Dirichlet beta reflection formula
Dirichlet beta function
8
Identity theorem
Complex analysis
19
Residue at infinity from an asymptotic constant
Complex analysis
27
Large-circle contour estimate
Complex analysis
23
Fuchsian differential equation
Complex analysis
392
16
Ordinary point criterion for a second-order equation
Fuchsian differential equation
18
Characteristic exponent at a regular singular point
Fuchsian differential equation
19
Regular singular point criterion for a second-order equation
Fuchsian differential equation
66
2
Regular singular point at infinity
Regular singular point criterion for a second-order equation
24
Logarithmic solution from a repeated Frobenius exponent
Regular singular point criterion for a second-order equation
25
Irregular singular point
Fuchsian differential equation
38
Papperitz symbol
Fuchsian differential equation
94
3
Fuchs relation
Papperitz symbol
18
Möbius transformation of a Papperitz symbol
Papperitz symbol
18
Dependent-variable rescaling of a Papperitz symbol
Papperitz symbol
37
Gauss hypergeometric equation
Fuchsian differential equation
142
5
Pfaff transformation
Gauss hypergeometric equation
33
Second local hypergeometric solution
Gauss hypergeometric equation
15
Hypergeometric connection formula at infinity
Gauss hypergeometric equation
42
Hypergeometric cancellation identity
Gauss hypergeometric equation
37
1
Laplace transform
Analysis
132
7
Laplace transform of a derivative
Laplace transform
14
Laplace transform shift theorem
Laplace transform
36
2
Laplace transform time-shift rule
Laplace transform shift theorem
9
Heaviside step function
Laplace transform shift theorem
20
Laplace transform of a periodic function
Laplace transform
10
Matrix-valued Laplace transform
Laplace transform
55
1
Laplace-transform solution of a constant-coefficient vector ODE
Matrix-valued Laplace transform
19
Hardy-Littlewood maximal function
Analysis
30
1
Hardy-Littlewood maximal inequality
Hardy-Littlewood maximal function
19
Watson lemma
Analysis
34
1
Reciprocal-amplitude Laplace integral
Watson lemma
14
Incomplete gamma function
Analysis
105
5
Asymptotic expansion by repeated integration by parts
Incomplete gamma function
44
2
Large-lower-limit expansion of the incomplete gamma function
Asymptotic expansion by repeated integration by parts
9
Fixed-lower-limit sine-integral expansion
Asymptotic expansion by repeated integration by parts
12
Fixed-lower-limit incomplete gamma asymptotic
Incomplete gamma function
22
Endpoint expansion of the incomplete gamma function
Incomplete gamma function
26
Stationary phase method
Analysis
103
3
Nondegenerate stationary point of a phase
Stationary phase method
24
One-dimensional stationary-phase formula
Stationary phase method
24
Nonstationary oscillatory integral
Stationary phase method
35
WKB approximation
Analysis
267
10
Airy turning-point connection formula
WKB approximation
37
Classical turning point
WKB approximation
29
WKB quantization condition
WKB approximation
105
3
Two-turning-point WKB quantization condition
WKB quantization condition
67
2
Semiclassical action integral
Two-turning-point WKB quantization condition
52
1
Removal of the first derivative from a second-order differential equation
WKB approximation
11
Irregular singular point at infinity
WKB approximation
33
Liouville-Green exponential ansatz
WKB approximation
34
1
Riccati equation
Liouville-Green exponential ansatz
12
Numerical analysis
Analysis
3k
119
Arithmetic operation
Numerical analysis
23
Schur stability criterion
Numerical analysis
32
Lie-Trotter splitting commutator error
Numerical analysis
22
Composite midpoint rule error
Numerical analysis
49
1
Quadratic substitution for a square-root endpoint singularity
Composite midpoint rule error
28
Polynomial interpolation
Numerical analysis
74
3
Lagrange interpolation polynomial
Polynomial interpolation
10
Divided difference
Polynomial interpolation
43
1
Newton interpolation polynomial
Divided difference
19
Discrete Fourier transform
Numerical analysis
144
6
Cooley--Tukey fast Fourier transform
Discrete Fourier transform
48
1
Even--odd fast Fourier transform recursion
Cooley--Tukey fast Fourier transform
17
Discrete cosine transform
Discrete Fourier transform
48
1
Discrete cosine transform from an even-reflected discrete Fourier transform
Discrete cosine transform
39
Discrete sine transform
Discrete Fourier transform
37
1
Discrete sine transform from a sign-modulated discrete cosine transform
Discrete sine transform
29
Gershgorin circle theorem
Numerical analysis
65
1
Gershgorin stability bound for a variable-coefficient diffusion stencil
Gershgorin circle theorem
50
Finite difference method
Numerical analysis
423
17
Central finite difference
Finite difference method
11
Forward difference operator
Finite difference method
54
2
Discrete antiderivative
Forward difference operator
17
Newton series for an integer-valued polynomial sequence
Forward difference operator
23
Von Neumann stability analysis
Finite difference method
145
5
Amplification factor
Von Neumann stability analysis
46
1
Amplification polynomial of a multilevel finite difference scheme
Von Neumann stability analysis
23
Crank-Nicolson diffusion scheme
Von Neumann stability analysis
22
Centered three-level wave scheme
Von Neumann stability analysis
20
Courant number
Finite difference method
24
Dirichlet discrete Laplacian
Finite difference method
173
5
Crank-Nicolson stability on a finite Dirichlet interval
Dirichlet discrete Laplacian
28
Five-point Dirichlet Laplacian as a Kronecker sum
Dirichlet discrete Laplacian
125
3
Thomas algorithm
Numerical analysis
22
Gaussian quadrature
Numerical analysis
19
Chebyshev polynomial
Numerical analysis
173
7
Chebyshev polynomial of the second kind
Chebyshev polynomial
28
Rational cosine of an integral submultiple of pi
Chebyshev polynomial
26
Positivity of Chebyshev derivatives beyond the unit interval
Chebyshev polynomial
22
Monic Chebyshev extremal polynomial
Chebyshev polynomial
22
Chebyshev polynomial domination lemma
Chebyshev polynomial
23
Chebyshev differential equation
Chebyshev polynomial
38
1
Chebyshev derivative Sturm-Liouville pair
Chebyshev differential equation
24
Backward differentiation formula
Numerical analysis
Constrained optimization
Numerical analysis
Equally spaced interpolation
Numerical analysis
Ford-Fulkerson algorithm
Numerical analysis
Givens rotation
Numerical analysis
51
1
QR decomposition by Givens rotations
Givens rotation
28
Gradient descent
Numerical analysis
216
9
Lipschitz gradient
Gradient descent
12
Exact line search for a positive-definite quadratic
Gradient descent
25
Conjugate gradient method
Gradient descent
75
2
Krylov subspace
Conjugate gradient method
13
Finite termination of the conjugate gradient method
Conjugate gradient method
32
Heavy-ball method
Gradient descent
86
3
Heavy-ball residual recurrence
Heavy-ball method
26
Heavy-ball error propagation matrix
Heavy-ball method
50
1
Peano kernel theorem
Numerical analysis
56
2
Four-point one-sided second-derivative formula
Peano kernel theorem
36
1
Sharp Peano-kernel constant for the four-point endpoint second derivative
Four-point one-sided second-derivative formula
27
Periodic trapezoidal Fourier aliasing
Numerical analysis
36
Fourier-Galerkin matrix for a drift-diffusion equation
Numerical analysis
51
Fourier spectral method for variable-coefficient advection
Numerical analysis
98
3
Positive Fourier-symbol Toeplitz matrix
Fourier spectral method for variable-coefficient advection
25
Real spectrum of a positive-Hermitian times Hermitian product
Fourier spectral method for variable-coefficient advection
26
Explicit Euler instability on a nonzero imaginary eigenvalue
Fourier spectral method for variable-coefficient advection
25
Runge-Kutta method
Numerical analysis
84
4
Local truncation error
Runge-Kutta method
37
Implicit Runge-Kutta method
Runge-Kutta method
31
1
Trapezoidal rule
Implicit Runge-Kutta method
16
Order of a Runge-Kutta method
Runge-Kutta method
16
Simplex algorithm
Numerical analysis
Sparse optimization
Numerical analysis
Stability function
Numerical analysis
Multistep method
Numerical analysis
61
3
Zero-stability
Multistep method
35
1
Root condition for a multistep method
Zero-stability
15
Dahlquist equivalence theorem
Multistep method
12
Numerical linear algebra
Numerical analysis
484
18
Gaussian elimination
Numerical linear algebra
20
LDL decomposition
Numerical linear algebra
42
Machine precision
Numerical linear algebra
12
Eigenpair residual
Numerical linear algebra
39
1
Backward error of an approximate eigenpair
Eigenpair residual
26
Householder QR decomposition
Numerical linear algebra
19
Householder tridiagonalization
Numerical linear algebra
12
Unshifted QR algorithm
Numerical linear algebra
219
5
Accumulated QR factorization identity
Unshifted QR algorithm
53
Symmetric bandwidth preservation under QR iteration
Unshifted QR algorithm
33
Simultaneous iteration interpretation of the QR algorithm
Unshifted QR algorithm
87
1
Block deflation in the QR algorithm
Unshifted QR algorithm
26
Stationary iterative method for a linear system
Numerical linear algebra
111
4
Matrix splitting
Stationary iterative method for a linear system
40
1
Jacobi method
Stationary iterative method for a linear system
45
1
Linear stability domain
Numerical analysis
186
5
Explicit Euler method
Linear stability domain
12
A-stability
Linear stability domain
51
1
A-stability of a symmetric two-stage implicit Runge-Kutta family
A-stability
38
Stiff two-mode linear system
Linear stability domain
47
Milne device for forward and backward Euler
Linear stability domain
43
Orthogonal polynomial
Numerical analysis
69
4
Monic orthogonal polynomial
Orthogonal polynomial
23
Hermite polynomial
Orthogonal polynomial
Zeros of orthogonal polynomials
Orthogonal polynomial
17
Jacobi matrix
Orthogonal polynomial
16
Quadrature rule
Numerical analysis
179
5
Nodal-polynomial criterion for quadrature exactness
Quadrature rule
30
Degree ceiling for quadrature exactness
Quadrature rule
32
Positivity of quadrature weights from degree 2n exactness
Quadrature rule
39
Convergence of positive quadrature rules
Quadrature rule
41
Moment matching
Quadrature rule
22
Asymptotic expansion
Analysis
263
9
Asymptotic sequence
Asymptotic expansion
107
3
Harmonic-mean refinement of an asymptotic sequence
Asymptotic sequence
23
Geometric-mean refinement of an asymptotic sequence
Asymptotic sequence
16
Termwise equivalent asymptotic scales need not preserve expansions
Asymptotic sequence
36
Laplace's method
Asymptotic expansion
65
1
Laplace asymptotic for the derivative of the Gamma function
Laplace's method
28
Method of steepest descent
Asymptotic expansion
91
2
Saddle point
Method of steepest descent
33
Simple-saddle contribution in steepest descent
Method of steepest descent
25
Bessel function
Analysis
210
7
Modified Bessel function
Bessel function
210
6
Laplace integral method for a differential equation
Modified Bessel function
185
5
Laplace integral solution of a singular third-order equation
Laplace integral method for a differential equation
36
Laguerre contour-integral amplitude
Laplace integral method for a differential equation
103
2
Real integral representation of the modified Bessel function I0
Laplace integral method for a differential equation
8
Branch cut
Analysis
99
4
Monodromy of a meromorphic integral
Branch cut
48
1
Residue contribution to path dependence
Monodromy of a meromorphic integral
22
Branch cut joining cancelling poles
Branch cut
27
Path independence by vanishing periods
Branch cut
24
Cauchy integral formula
Analysis
24
1
Cauchy estimate
Cauchy integral formula
24
Cauchy-Riemann equations
Analysis
Complex logarithm
Analysis
82
3
Complex exponentiation
Complex logarithm
16
Analytic logarithm on the positive-axis slit plane
Complex logarithm
66
1
Positive-axis keyhole beta integral
Analytic logarithm on the positive-axis slit plane
38
Critical point
Analysis
51
2
Critical set
Critical point
19
Local minimum
Critical point
32
Damped harmonic oscillator
Analysis
12
1
Forced harmonic oscillator
Damped harmonic oscillator
12
Differentiable function
Analysis
46
2
Smooth function
Differentiable function
9
Existence of partial derivatives does not imply their continuity
Differentiable function
37
Differentiation under the integral sign
Analysis
58
1
Second-order Hadamard lemma
Differentiation under the integral sign
11
Dirichlet problem
Analysis
Euler-Lagrange equation
Analysis
126
5
Beltrami identity
Euler-Lagrange equation
17
Catenary
Euler-Lagrange equation
25
Natural boundary conditions for a free endpoint
Euler-Lagrange equation
60
1
Free-endpoint normal mode of a coupled variational functional
Natural boundary conditions for a free endpoint
30
Euler-Lagrange equations for two fields
Euler-Lagrange equation
14
Exponential decay
Analysis
First-step analysis
Analysis
23
Forced oscillator
Analysis
20
1
Steady-state response
Forced oscillator
20
Fourier series
Analysis
88
5
Leibniz formula for pi
Fourier series
10
Trigonometric polynomial
Fourier series
29
Fourier coefficient
Fourier series
15
Fourier cosine series
Fourier series
19
Fourier series of x cubed minus pi squared x
Fourier series
15
Fourier transform
Analysis
126
5
Scaling property of the Fourier transform
Fourier transform
56
1
Scaling necessity for an Lp to Lq Fourier bound
Scaling property of the Fourier transform
34
Fourier transform of inverse distance in three dimensions
Fourier transform
27
Hilbert transform
Fourier transform
13
Sinc-squared integral
Fourier transform
16
Gradient flow
Analysis
Green function
Analysis
111
5
One-dimensional modified Helmholtz Green function
Green function
33
1
Dirichlet half-line Green function for d2 minus 1
One-dimensional modified Helmholtz Green function
12
Neumann Green function for a second-order ordinary differential equation
Green function
44
Causal Green function
Green function
19
One-dimensional outgoing Green function
Green function
15
Hermite differential equation
Analysis
Higher-order Euler-Lagrange equation
Analysis
63
2
Clamped-free beam under uniform load and endpoint force
Higher-order Euler-Lagrange equation
49
1
Endpoint force derivative of clamped-free beam energy
Clamped-free beam under uniform load and endpoint force
22
Laurent series
Analysis
30
1
Laurent theorem
Laurent series
30
Method of characteristics
Analysis
Differential equation
Analysis
980
51
Ordinary differential equation
Differential equation
980
50
Linear differential equation
Ordinary differential equation
39
1
Inhomogeneous linear differential equation
Linear differential equation
26
Local existence for a scalar autonomous ordinary differential equation with continuous vector field
Ordinary differential equation
30
Singular perturbation
Ordinary differential equation
30
Lie point symmetry of an ordinary differential equation
Ordinary differential equation
91
3
Prolongation of a vector field
Lie point symmetry of an ordinary differential equation
65
2
Boundary value problem
Ordinary differential equation
19
Boundary condition
Ordinary differential equation
69
3
Dirichlet boundary condition
Boundary condition
14
Neumann boundary condition
Boundary condition
16
Far-field boundary condition
Boundary condition
20
Sign-decay differential equation
Ordinary differential equation
93
1
Explicit Euler method for the sign-decay equation
Sign-decay differential equation
47
Parameter sensitivity equation
Ordinary differential equation
29
Differential inequality
Ordinary differential equation
20
Bernoulli differential equation
Ordinary differential equation
11
Tangent addition functional equation
Ordinary differential equation
21
Linear ordinary differential equation
Ordinary differential equation
511
29
Homogeneous solution
Linear ordinary differential equation
25
Particular solution
Linear ordinary differential equation
26
Dominant eigenmode in a forced linear system
Linear ordinary differential equation
33
Second-order linear differential equation
Linear ordinary differential equation
256
16
Integrating factor
Linear ordinary differential equation
43
1
Resonance in a differential equation
Linear ordinary differential equation
35
1
Linear system of differential equations
Linear ordinary differential equation
57
3
Jump condition for an impulse
Linear ordinary differential equation
22
First integral
Ordinary differential equation
17
Real analytic function
Analysis
Residue
Analysis
33
1
Residue at a pole of order n
Residue
11
Residue theorem
Analysis
118
3
Upper-half-plane indentation rule
Residue theorem
55
Residue cancellation by symmetry
Residue theorem
15
Symmetric residue summation
Residue theorem
23
Schwarz lemma
Analysis
Spectral radius
Analysis
14
Spherical average
Analysis
Sturm-Liouville theory
Analysis
177
7
Sturm-Liouville problem
Sturm-Liouville theory
78
2
Dirichlet eigenvalue
Sturm-Liouville problem
25
Sturm-Liouville eigenfunction expansion
Sturm-Liouville problem
27
Self-adjoint differential operator
Sturm-Liouville theory
99
3
Self-adjoint differential equation
Self-adjoint differential operator
27
Solvability condition at a Sturm-Liouville eigenvalue
Self-adjoint differential operator
55
1
Leading nonlinear eigenvalue shift in a Sturm-Liouville problem
Solvability condition at a Sturm-Liouville eigenvalue
22
Travelling wave
Analysis
88
4
Travelling-wave reduction of a reaction-diffusion system
Travelling wave
22
Linear spreading speed
Travelling wave
66
2
Fisher-KPP minimum wave speed
Linear spreading speed
19
Pulled travelling front
Linear spreading speed
22
Uniqueness of Poisson equation
Analysis
Uniqueness theorem for ordinary differential equations
Analysis
Real analysis
Analysis
5k
255
Lipschitz continuity
Real analysis
44
1
Lipschitz bound
Lipschitz continuity
22
Big O notation
Real analysis
17
Real line
Real analysis
46
2
Real interval
Real line
27
1
Closed real interval
Real interval
13
Extreme value theorem
Real analysis
12
Coercive function
Real analysis
40
Lp norm
Real analysis
24
Sequence and series
Real analysis
1k
54
Sequence
Sequence and series
83
4
Bounded sequence
Sequence
20
Binary sequence
Sequence
15
Monotone sequence
Sequence
14
Limit superior
Sequence
22
Series
Sequence and series
36
1
Partial sum
Series
22
Fekete lemma
Sequence and series
24
Bolzano-Weierstrass theorem
Sequence and series
120
2
Nested interval theorem
Bolzano-Weierstrass theorem
26
Unique subsequential limit of a bounded sequence
Bolzano-Weierstrass theorem
49
Convergent sequence
Sequence and series
70
3
Cesaro mean
Convergent sequence
28
Geometric mean of a sequence
Convergent sequence
17
Monotone bounded sequence
Convergent sequence
25
Fibonacci number
Sequence and series
24
1
Fibonacci determinant identity
Fibonacci number
13
Pointwise convergence
Sequence and series
Power series
Sequence and series
138
7
Radius of convergence
Power series
80
3
Cauchy product
Power series
21
Termwise differentiation of a power series
Power series
22
Leading Taylor term
Power series
15
Uniform convergence
Sequence and series
101
3
Uniformly Cauchy sequence
Uniform convergence
19
Locally uniform convergence
Uniform convergence
58
1
Geometric series
Sequence and series
10
1
Finite geometric series
Geometric series
10
Cauchy sequence
Sequence and series
27
1
Completeness of the real numbers
Cauchy sequence
11
Conditional convergence
Sequence and series
102
3
Alternating series test
Conditional convergence
58
1
Rearrangement of a series
Conditional convergence
31
P-series
Sequence and series
8
Absolute convergence
Sequence and series
63
3
Squares of a summable positive sequence
Absolute convergence
17
Geometric means of two summable positive sequences
Absolute convergence
5
Weighted square roots of a summable sequence
Absolute convergence
26
Total variation
Sequence and series
16
Harmonic sum
Sequence and series
7
Asymptotic equivalence
Sequence and series
34
2
Asymptotic comparison of exponential functions
Asymptotic equivalence
12
Stirling formula
Asymptotic equivalence
15
Convergent series
Sequence and series
113
4
Limit comparison test
Convergent series
23
Ratio test
Convergent series
76
2
Uniform limit
Sequence and series
17
Calculus
Real analysis
2k
106
Floor function
Calculus
14
Integral
Calculus
117
6
Double integral
Integral
21
Monotonicity of the Lebesgue integral
Integral
18
Antiderivative
Integral
26
1
Gaussian integral
Integral
34
1
Taylor polynomial
Calculus
Exponential function
Calculus
68
6
Gaussian function
Exponential function
22
Natural logarithm
Exponential function
14
Hyperbolic cosine
Exponential function
7
Hyperbolic sine
Exponential function
7
Hyperbolic cotangent
Exponential function
12
Hyperbolic secant
Exponential function
6
Fundamental theorem of calculus
Calculus
Mean value theorem
Calculus
Rolle theorem
Calculus
Taylor theorem
Calculus
31
2
Taylor theorem with Lagrange remainder
Taylor theorem
20
Taylor series
Taylor theorem
11
Limit of a function
Calculus
229
10
Continuous function
Limit of a function
196
8
Squeeze theorem
Limit of a function
18
Derivative
Calculus
119
8
Darboux theorem
Derivative
12
Product rule
Derivative
37
2
Quotient rule
Derivative
13
Chain rule
Derivative
39
2
Monotone function
Calculus
46
2
Strictly increasing function
Monotone function
22
Lebesgue theorem on differentiability of monotone functions
Monotone function
13
Multivariable calculus
Calculus
856
53
Partial derivative
Multivariable calculus
43
2
Hessian matrix
Multivariable calculus
52
1
Vector calculus
Multivariable calculus
366
25
Jacobian matrix
Multivariable calculus
32
1
Change of variables formula
Multivariable calculus
118
8
Differentiable map
Multivariable calculus
234
10
Symmetry in integration
Calculus
16
Integration by parts
Calculus
12
Hyperbolic tangent
Calculus
38
2
Small-argument expansion of the hyperbolic tangent
Hyperbolic tangent
13
Hyperbolic Pythagorean identity
Hyperbolic tangent
7
Even function
Calculus
12
Odd function
Calculus
12
Riemann integration
Real analysis
225
9
Continuous approximation of a Riemann-integrable function
Riemann integration
23
Riemann integral
Riemann integration
33
1
Monotonicity of the Riemann integral
Riemann integral
33
Riemann integrability criterion
Riemann integration
53
1
Continuous functions are Riemann integrable
Riemann integrability criterion
36
Darboux sum
Riemann integration
38
1
Upper and lower Darboux integrals
Darboux sum
17
Improper integration by truncation
Riemann integration
66
1
First occurrence of a decimal digit
Improper integration by truncation
37
Supremum
Real analysis
13
Convex function
Real analysis
198
10
Pointwise maximum of convex functions
Convex function
17
Softplus function
Convex function
8
Absolute value function
Convex function
12
Strongly convex function
Convex function
12
Strictly convex function
Convex function
43
1
Uniqueness of a minimizer of a strictly convex function
Strictly convex function
27
Convexity domain of x cubed plus y cubed plus Axy
Convex function
37
Perspective function
Convex function
13
Subgradient inequality
Convex function
15
Jensen inequality
Convex function
20
Concave function
Real analysis
79
2
Strictly concave function
Concave function
62
1
Product maximizer on a compact convex subset of the positive orthant
Strictly concave function
28
Measure theory
Real analysis
1k
58
Measure density
Measure theory
112
4
Lebesgue differentiation theorem
Measure density
62
2
Lebesgue density theorem
Measure density
25
Lebesgue measurable set
Measure theory
28
Almost everywhere
Measure theory
20
Lebesgue integrable function
Measure theory
11
Conditional expectation
Measure theory
61
1
Tower property of conditional expectation
Conditional expectation
24
Fatou lemma
Measure theory
56
2
Proof of Fatou lemma
Fatou lemma
25
Strict inequality in Fatou lemma
Fatou lemma
25
Ergodic theory
Measure theory
183
7
Measure-preserving transformation
Ergodic theory
107
4
Birkhoff ergodic theorem
Ergodic theory
65
1
Measure
Measure theory
163
7
Measure space
Measure
21
Pushforward measure
Measure
32
Change of measure
Measure
26
Haar measure
Measure
28
Countable additivity
Measure
12
Finite measure
Measure
10
Sigma-finite measure
Measure
16
Indicator function
Measure theory
33
1
Indicator vector
Indicator function
18
Sigma-algebra
Measure theory
94
4
Borel set
Sigma-algebra
12
Pi-system
Sigma-algebra
12
Dynkin system
Sigma-algebra
56
1
Fubini's theorem
Measure theory
20
Sigma-finite uniqueness theorem for measures
Measure theory
75
2
Lebesgue measure
Sigma-finite uniqueness theorem for measures
45
1
Lebesgue space
Measure theory
87
2
Riesz-Fischer theorem
Lebesgue space
14
Banach intersection of L1 and L2
Lebesgue space
41
Approximation by nonnegative simple functions
Measure theory
22
Monotone convergence theorem
Measure theory
18
Dominated convergence theorem
Measure theory
21
Tonelli theorem
Measure theory
16
Absolute continuity of measures
Measure theory
176
7
Lusin condition N
Absolute continuity of measures
17
Image-length measure of a strictly increasing continuous function
Absolute continuity of measures
32
Uniform absolute continuity for a finite measure
Absolute continuity of measures
46
Mutually absolutely continuous measures
Absolute continuity of measures
14
Radon-Nikodym theorem
Absolute continuity of measures
30
1
Dominating mixture measure
Absolute continuity of measures
18
Lebesgue-Stieltjes measure
Measure theory
29
Lp inclusion on a finite measure space
Measure theory
27
1
Power singularity integrability
Lp inclusion on a finite measure space
17
Partial differential equation
Analysis
2k
114
Modified Helmholtz equation
Partial differential equation
14
Poisson equation
Partial differential equation
32
2
Poisson kernel for the upper half-space
Poisson equation
32
1
Poisson integral
Poisson kernel for the upper half-space
21
Diffusion equation
Partial differential equation
720
29
Fick's first law
Diffusion equation
24
Heat equation
Diffusion equation
213
8
Sinusoidally forced heat equation on a half-line
Heat equation
27
Potential Burgers equation
Heat equation
22
Heat kernel
Heat equation
97
3
Duhamel principle
Heat equation
54
1
Advection-diffusion equation
Diffusion equation
31
1
Advection-diffusion heat-kernel solution
Advection-diffusion equation
9
Nonlinear diffusion equation
Diffusion equation
52
2
Two-dimensional Barenblatt solution with diffusivity proportional to concentration
Nonlinear diffusion equation
34
1
Reaction-diffusion system
Diffusion equation
388
13
Morphogen reaction-diffusion equation
Reaction-diffusion system
54
2
Fast-inhibitor elimination in a reaction-diffusion system
Reaction-diffusion system
52
1
Turing instability
Reaction-diffusion system
127
3
Brusselator
Reaction-diffusion system
143
3
Similarity solution
Partial differential equation
16
Separation of variables
Partial differential equation
31
1
Dirichlet problem on an annulus for one Fourier mode
Separation of variables
15
Polynomial ansatz
Partial differential equation
16
Harmonic function
Partial differential equation
377
14
Harmonic function as the real part of a holomorphic function
Harmonic function
28
Subharmonic function
Harmonic function
66
1
Maximum principle for subharmonic functions
Subharmonic function
38
Maximum principle for harmonic functions
Harmonic function
35
Harmonic Liouville theorem
Harmonic function
55
1
Surjectivity of a nonconstant entire harmonic function
Harmonic Liouville theorem
22
Scale-periodic harmonic function from an elliptic function
Harmonic function
48
Harmonic polynomial
Harmonic function
10
Harmonic conjugate
Harmonic function
85
2
Local harmonic conjugate
Harmonic conjugate
22
Log modulus has no global harmonic conjugate on the punctured plane
Harmonic conjugate
49
Harmonic functions are smooth
Harmonic function
18
Harmonic functions are real analytic
Harmonic function
23
1
Unique continuation for harmonic functions
Harmonic functions are real analytic
16
Smooth bump function
Partial differential equation
41
1
Disjoint-support zero-product construction
Smooth bump function
20
Laplace operator
Partial differential equation
122
8
Laplace equation
Laplace operator
49
2
Green function of the Laplacian
Laplace equation
16
Laplace equation in polar coordinates
Laplace equation
24
Radial Laplacian
Laplace operator
9
Laplacian in spherical coordinates
Laplace operator
27
1
Axisymmetric harmonic function
Laplacian in spherical coordinates
10
Laplacian eigenfunction
Laplace operator
26
1
Dirichlet Laplacian eigenfunction
Laplacian eigenfunction
14
Wave equation
Partial differential equation
681
36
Elastic wave
Wave equation
259
13
Helmholtz separation of elastic waves
Elastic wave
22
P-wave
Elastic wave
30
1
S-wave
Elastic wave
98
4
Elastic-interface boundary conditions
Elastic wave
46
2
Acoustic reflection and transmission at an interface
Elastic wave
43
1
Wave equation on a string
Wave equation
91
4
Linearly damped string
Wave equation on a string
72
3
Normal mode
Wave equation
38
2
Modal energy distribution
Normal mode
13
Node of a standing wave
Normal mode
13
Fourier transform method for the wave equation
Wave equation
14
D'Alembert formula
Wave equation
49
2
D'Alembert formula with initial velocity
D'Alembert formula
43
1
Wavenumber
Wave equation
18
Dispersion relation
Wave equation
130
5
Growth rate
Dispersion relation
25
Phase velocity and group velocity
Dispersion relation
83
3
Klein-Gordon equation
Wave equation
73
2
Stationary-phase asymptotic of a Klein-Gordon wave along a subluminal ray
Klein-Gordon equation
62
1
Green second identity
Partial differential equation
38
1
Green's third identity
Green second identity
22
Green's first identity
Partial differential equation
17
Lie point symmetry
Partial differential equation
233
9
Infinitesimal generator of a Lie point symmetry
Lie point symmetry
14
Prolongation of a Lie point symmetry
Lie point symmetry
58
1
Second prolongation of a Lie point symmetry
Prolongation of a Lie point symmetry
19
Scaling symmetry of a partial differential equation
Lie point symmetry
42
1
Simultaneous spacetime scaling symmetry of the wave equation
Scaling symmetry of a partial differential equation
26
Symmetry reduction of a partial differential equation
Lie point symmetry
97
3
Group-invariant solution
Symmetry reduction of a partial differential equation
31
Projective Lie symmetry of the potential Burgers equation
Symmetry reduction of a partial differential equation
36
Similarity variable
Symmetry reduction of a partial differential equation
12
Distribution theory
Analysis
67
4
Distributional identity
Distribution theory
19
Dirac delta function
Distribution theory
31
2
Distributional derivative of the Heaviside step function
Dirac delta function
12
Dirac delta scaling
Dirac delta function
9
Calculus of variations
Analysis
320
17
Functional
Calculus of variations
140
6
Energy functional
Functional
21
Lagrangian
Functional
45
1
Lagrangian function in constrained optimization
Lagrangian
15
Variation
Functional
57
2
First variation
Variation
39
1
Stationary point
Calculus of variations
16
Dirichlet principle
Calculus of variations
18
Optical ray in cylindrical coordinates
Calculus of variations
26
1
Helical extremal
Optical ray in cylindrical coordinates
14
Noether theorem
Calculus of variations
11
Second variation
Calculus of variations
99
4
Jacobi equation
Second variation
21
Conjugate point
Second variation
19
Wirtinger inequality
Second variation
37
1
Periodic Wirtinger inequality
Wirtinger inequality
24
Fixed-point theorem
Analysis
263
8
Knaster-Tarski theorem
Fixed-point theorem
37
Fixed-point property
Fixed-point theorem
74
2
Homeomorphism invariance of the fixed-point property
Fixed-point property
38
Fixed-point property of a closed interval
Fixed-point property
21
Contraction mapping theorem
Fixed-point theorem
136
3
Continuous dependence of the fixed point of a uniform contraction
Contraction mapping theorem
46
Iterated contraction
Contraction mapping theorem
30
Local contraction proof for Newton iteration
Contraction mapping theorem
41
Uniform approximation
Analysis
61
2
Best uniform approximation
Uniform approximation
23
Chebyshev alternation theorem
Uniform approximation
25
Ancestors
(3)
Area of mathematics
Mathematics
Home
Codex Wiki