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Words: 4k
Articles: 182
Table of contents
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
Great Picard theorem
Little Picard theorem
18
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
Sum-and-ratio substitution for gamma integrals
Beta--gamma identity
20
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
Evaluation map on a space of germs
Space of germs of holomorphic functions
24
Germ surface of the square root of z to the eighth minus one
Space of germs of holomorphic functions
35
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 the derivative of a rational function
Degree of a rational map of the Riemann sphere
72
Degree of an elliptic function
Valency theorem
76
1
Degree of the derivative of an elliptic function
Degree of an elliptic function
38
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
Finite conformal quotient of a Riemann surface
Local cyclic quotient of a Riemann surface
45
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
Cylinder as a punctured plane
Punctured complex plane
11
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
Riemann removable singularity theorem
Removable singularity
13
Removable singularity at infinity
Removable singularity
17
Uniform L2 circle bound for a removable singularity
Removable singularity
35
Pole
Classification of isolated singularities
169
5
Meromorphic function
Pole
117
3
Rational function
Meromorphic function
92
2
Partial fraction decomposition
Rational function
37
Order of vanishing
Rational function
32
Exponential of a pole is an essential singularity
Pole
32
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
Maximum modulus principle from the complex open mapping theorem
Open mapping theorem
30
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
Elementary specialization of a hypergeometric solution
Hypergeometric cancellation identity
23
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Analysis
Area of mathematics
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