List of Figures
-
1 Algebraic and Analytic Methods
-
2 Asymptotic Approximations
-
3 Numerical Methods
-
4 Elementary Functions
-
5 Gamma Function
-
6 Exponential, Logarithmic, Sine, and Cosine Integrals
-
6.3.1 , , .
-
6.3.2 , .
-
6.3.3 , , .
-
6.16.1 Gibbs phenomenon.
-
6.16.2 , , .
-
7 Error Functions, Dawson’s and Fresnel Integrals
-
7.3.1 , , .
-
7.3.2 , .
-
7.3.3 , , .
-
7.3.4 , .
-
7.3.5 , , .
-
7.3.6 , , .
-
7.18.1 Repeated integrals of the scaled complementary error function.
-
7.19.1 Voigt function ,
, , , .
-
7.19.2 Voigt function ,
, , , .
-
7.20.1 Cornu’s spiral.
-
8 Incomplete Gamma and Related Functions
-
8.3.1 , = 0.25, 1, 2, 2.5, 3.
-
8.3.2 ,
= 0.25, 0.5, 0.75, 1.
-
8.3.3 ,
= 1, 2, 2.5, 3.
-
8.3.4 (= ),
= 0.25, 0.5, 0.75, 1, 2.
-
8.3.5 (= ),
= 0.25, 0.5, 1, 2.
-
8.3.6 (= ),
, .
-
8.3.7 (= ),
, .
-
8.3.8 ,
, .
-
8.3.9 ,
, .
-
8.3.10 ,
, .
-
8.3.11 ,
, .
-
8.3.12 ,
, .
-
8.3.13 ,
, .
-
8.3.14 ,
, .
-
8.3.15 ,
, .
-
8.3.16 ,
, .
-
8.19.1 ,
, .
-
8.19.2 ,
, .
-
8.19.3 ,
, .
-
8.19.4 ,
, .
-
8.19.5 ,
, .
-
9 Airy and Related Functions
-
9.3.1 , , .
-
9.3.2 , , .
-
9.3.3 .
-
9.3.4 .
-
9.3.5 .
-
9.3.6 .
-
9.12.1 , .
-
9.12.2 , .
-
9.13.1 Paths , ,
, .
-
9.13.2 Paths , ,
.
-
10 Bessel Functions
-
10.3.1 , , , , .
-
10.3.2 , , , .
-
10.3.3 , , , .
-
10.3.4 , , .
-
10.3.5 , , .
-
10.3.6 , , .
-
10.3.7 , , .
-
10.3.8 , , .
-
10.3.9 , , .
-
10.3.10 , , .
-
10.3.11 , , .
-
10.3.12 , , .
-
10.3.13 , , .
-
10.3.14 , , .
-
10.3.15 , , .
-
10.3.16 , , .
-
10.3.17 , , .
-
10.3.18 , , .
-
10.3.19 , , .
-
10.20.1 -plane.
-
10.20.2 -plane.
-
10.20.3 Domain .
-
10.21.1 Zeros of in .
-
10.21.2 Zeros of in .
-
10.21.3 Zeros of in .
-
10.21.4 Zeros of in .
-
10.21.5 Zeros of in .
-
10.21.6 Zeros of in .
-
10.23.1 Graf’s and Gegenbauer’s addition theorems.
-
10.26.1 , , , , .
-
10.26.2 , , , , .
-
10.26.3 , , .
-
10.26.4 , , .
-
10.26.5 , , .
-
10.26.6 , , .
-
10.26.7 , , .
-
10.26.8 , , .
-
10.26.9 , , .
-
10.26.10 , .
-
10.41.1 -plane.
-
10.41.2 -plane.
-
10.48.1 , , .
-
10.48.2 , , .
-
10.48.3 , ,
,
.
-
10.48.4 , ,
,
.
-
10.48.5 , , ,
.
-
10.48.6 , .
-
10.48.7 , , ,
.
-
10.62.1 , , ,
, .
-
10.62.2 , , ,
, .
-
10.62.3 ,
,
, .
-
10.62.4 , ,
, .
-
11 Struve and Related Functions
-
11.3.1 , , .
-
11.3.2 , , .
-
11.3.3 , , .
-
11.3.4 , , .
-
11.3.5 , , .
-
11.3.6 , , .
-
11.3.7 , , .
-
11.3.8 , , .
-
11.3.9 , , .
-
11.3.10 , , .
-
11.3.11 , , .
-
11.3.12 , , .
-
11.3.13 , , .
-
11.3.14 , , .
-
11.3.15 , , .
-
11.3.16 , , .
-
11.3.17 , , .
-
11.3.18 , , .
-
11.3.19 , , .
-
11.3.20 , , .
-
11.10.1 , , .
-
11.10.2 , , .
-
11.10.3 , , .
-
11.10.4 , , .
-
12 Parabolic Cylinder Functions
-
12.3.1 , = 0.5, 2, 3.5, 5, 8.
-
12.3.2 , = 0.5, 2, 3.5, 5, 8.
-
12.3.3 , = , , , .
-
12.3.4 , = , , , .
-
12.3.5 , , , .
-
12.3.6 , , , .
-
12.3.7 , , .
-
12.3.8 , , .
-
12.3.9 , , .
-
12.3.10 , , .
-
12.14.1 , , , .
-
12.14.2 , , , .
-
12.14.3 , , , .
-
12.14.4 , , .
-
13 Confluent Hypergeometric Functions
-
14 Legendre and Related Functions
-
14.4.1 , .
-
14.4.2 , .
-
14.4.3 , .
-
14.4.4 , .
-
14.4.5 , .
-
14.4.6 , .
-
14.4.7 , .
-
14.4.8 , .
-
14.4.9 , .
-
14.4.10 , .
-
14.4.11 , .
-
14.4.12 , .
-
14.4.13 , .
-
14.4.14 , .
-
14.4.15 , .
-
14.4.16 , .
-
14.4.17 , .
-
14.4.18 , .
-
14.4.19 , .
-
14.4.20 , .
-
14.4.21 , .
-
14.4.22 , .
-
14.4.23 , .
-
14.4.24 , .
-
14.4.25 , .
-
14.4.26 , .
-
14.4.27 , .
-
14.4.28 , .
-
14.4.29 , , .
-
14.4.30 , , .
-
14.4.31 , , .
-
14.4.32 , , .
-
14.20.1 , .
-
14.20.2 , .
-
14.20.3 , .
-
14.20.4 , .
-
14.20.5 , .
-
14.20.6 , .
-
14.20.7 .
-
14.20.8 , .
-
14.22.1 , , .
-
14.22.2 ,
, .
-
14.22.3 ,
, .
-
14.22.4 ,
, .
-
15 Hypergeometric Function
-
15.3.1 .
-
15.3.2 .
-
15.3.3 .
-
15.3.4 .
-
15.3.5 .
-
15.3.6 .
-
15.3.7 .
-
15.6.1 Contour of integration in (15.6.5).
-
16 Generalized Hypergeometric Functions & Meijer G-Function
-
18 Orthogonal Polynomials
-
18.4.1 Jacobi polynomials , .
-
18.4.2 Jacobi polynomials , .
-
18.4.3 Chebyshev polynomials , .
-
18.4.4 Legendre polynomials , .
-
18.4.5 Laguerre polynomials , .
-
18.4.6 Laguerre polynomials ,
.
-
18.4.7 Monic Hermite polynomials
, .
-
18.4.8 Laguerre polynomials ,
, .
-
18.4.9 Laguerre polynomials ,
, .
-
18.21.1 Askey scheme.
-
19 Elliptic Integrals
-
19.3.1 , , .
-
19.3.2 , , .
-
19.3.3 , , .
-
19.3.4 , , .
-
19.3.5 , , .
-
19.3.6 , , .
-
19.3.7 , ,
.
-
19.3.8 , ,
.
-
19.3.9 , ,
.
-
19.3.10 , ,
.
-
19.3.11 , ,
.
-
19.3.12 , ,
.
-
19.17.1 , , .
-
19.17.2 , , .
-
19.17.3 , , .
-
19.17.4 , , .
-
19.17.5 , , .
-
19.17.6 , , .
-
19.17.7 , , .
-
19.17.8 , , .
-
20 Theta Functions
-
20.2.1 Fundamental parallelogram.
-
20.3.1 ,
,
.
-
20.3.2 ,
,
= 0.05, 0.5, 0.7, 0.9.
-
20.3.3 ,
,
= 0.05, 0.5, 0.7, 0.9.
-
20.3.4 ,
,
= 0.05, 0.5, 0.7, 0.9.
-
20.3.5 ,
,
= 0.05, 0.5, 0.7, 0.9.
-
20.3.6 ,
,
= 0, 0.4, 5, 10, 40.
-
20.3.7 ,
,
= 0, 0.4, 5, 10, 40.
-
20.3.8 ,
,
= 0, 0.4, 5, 10, 40.
-
20.3.9 ,
,
= 0, 0.4, 5, 10, 40.
-
20.3.10 ,
,
.
-
20.3.11 ,
,
.
-
20.3.12 ,
,
.
-
20.3.13 ,
,
.
-
20.3.14 ,
,
.
-
20.3.15 ,
,
.
-
20.3.16 ,
,
.
-
20.3.17 ,
,
.
-
20.3.18 ,
,
.
-
20.3.19 ,
,
.
-
20.3.20 ,
,
.
-
20.3.21 ,
,
.
-
21 Multidimensional Theta Functions
-
(a)
(b)
(c)
(a)
(b)
(c)
(a)
(b)
(c)
21.4.1
parametrized by (21.4.1).
-
21.4.2 ,
, .
-
21.4.3 ,
, .
-
21.4.4 ,
, .
-
21.4.5 ,
, .
-
21.7.1 A basis of cycles for a genus 2 surface.
-
21.9.1 Two-dimensional periodic waves in a shallow water wave tank.
-
21.9.2 Contour plot of a two-phase solution of Equation (21.9.3).
-
22 Jacobian Elliptic Functions
-
22.3.1 , , ,
,
,
.
-
22.3.2 , , ,
,
,
.
-
22.3.3 , , ,
,
,
.
-
22.3.4 , , ,
,
,
.
-
22.3.5 , , ,
,
,
.
-
22.3.6 , , ,
,
,
.
-
22.3.7 , , ,
,
,
.
-
22.3.8 , , ,
,
,
.
-
22.3.9 , , ,
,
,
.
-
22.3.10 , , ,
,
,
.
-
22.3.11 , , ,
,
,
.
-
22.3.12 , , ,
,
,
.
-
22.3.13
for ,
to 20,
.
-
22.3.14
for ,
to 20,
.
-
22.3.15
for ,
to 20,
.
-
22.3.16 , , , .
-
22.3.17 , , , .
-
22.3.18 , , , .
-
22.3.19 , , , .
-
22.3.20 , , , .
-
22.3.21 , , .
-
22.3.22 .
-
22.3.23 .
-
22.3.24 .
-
22.3.25 .
-
22.3.26 Density plot of .
-
22.3.27 Density plot of .
-
22.3.28 Density plot of .
-
22.3.29 Density plot of .
-
(a)
(b)
(c)
22.4.1 Poles, zeros of the principal Jacobian elliptic functions.
-
22.4.2 Fundamental unit cell.
-
22.16.1 , , .
-
22.16.2 , , .
-
22.16.3 , , .
-
22.19.1 , , .
-
23 Weierstrass Elliptic and Modular Functions
-
23.4.1 , ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.2 , ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.3 , ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.4 , ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.5 , ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.6 , ,
= 0.1, 0.2, 0.5, 0.8.
-
23.4.7 , , = 0.2, 0.8, 0.95, 0.99.
-
23.4.8 ,
,
,
.
-
23.4.9 , ,
-
23.4.10 , ,
-
23.4.11 , ,
-
23.4.12 , , .
-
23.5.1 Rhombic lattice. .
-
23.5.2 Equianharmonic lattice.
,
.
-
23.16.1 , , ,
.
-
23.16.2 , , .
-
23.16.3 , , .
-
24 Bernoulli and Euler Polynomials
-
25 Zeta and Related Functions
-
25.3.1 , , .
-
25.3.2 , , .
-
25.3.3 , , .
-
25.3.4 , .
-
25.3.5 , .
-
25.3.6 , .
-
25.11.1 , = 0.3, 0.5, 0.8, 1, .
-
25.11.2 , , .
-
25.12.1 ,
-
25.12.2 ,
,
.
-
26 Combinatorial Analysis
-
28 Mathieu Functions and Hill’s Equation
-
28.2.1 Eigenvalues ,
of Mathieu’s equation.
-
28.3.1 , , .
-
28.3.2 , , .
-
28.3.3 , , .
-
28.3.4 , , .
-
28.3.5 , , .
-
28.3.6 , , .
-
28.3.7 , , .
-
28.3.8 , , .
-
28.3.9 , , .
-
28.3.10 , , .
-
28.3.11 , , .
-
28.3.12 , , .
-
28.3.13 , , .
-
28.5.1 , , .
-
28.5.2 , , .
-
28.5.3 , , .
-
28.5.4 , , .
-
28.5.5 , , .
-
28.5.6 , , .
-
28.7.1 Branch point of the eigenvalues and
:
.
-
28.13.1 , ;
,
(’s), (’s).
-
28.13.2 , , .
-
28.13.3 , , .
-
28.13.4 , , .
-
28.13.5 , , .
-
28.17.1 Stability chart for eigenvalues of Mathieu’s equation (28.2.1).
-
28.21.1 , , .
-
28.21.2 , , .
-
28.21.3 , .
-
28.21.4 , , .
-
28.21.5 , , .
-
28.21.6 , , .
-
29 Lamé Functions
-
29.2.1 Singularities of Lamé’s equation.
-
29.4.1 , , .
-
29.4.2 .
-
29.4.3 , .
-
29.4.4 , , .
-
29.4.5 , , .
-
29.4.6 .
-
29.4.7
-
29.4.8 , .
-
29.4.9 .
-
29.4.10 .
-
29.4.11 .
-
29.4.12 .
-
29.4.13 ,
, .
-
29.4.14 ,
, .
-
29.4.15 ,
, .
-
29.4.16 ,
, .
-
29.4.17 ,
, .
-
29.4.18 ,
, .
-
29.4.19 ,
, .
-
29.4.20 ,
, .
-
29.4.21 ,
, .
-
29.4.22 ,
, .
-
29.4.23 ,
, .
-
29.4.24 ,
, .
-
29.4.25 .
-
29.4.26 .
-
29.4.27 .
-
29.4.28 .
-
29.4.29 .
-
29.4.30 .
-
29.4.31 .
-
29.4.32 .
-
29.13.1 , .
-
29.13.2 , .
-
29.13.3 , .
-
29.13.4 , .
-
29.13.5 for
, .
.
-
29.13.6 for
, .
.
-
29.13.7 for
, .
.
-
29.13.8 for
, .
.
-
29.13.9 for
, .
.
-
29.13.10 for
, .
.
-
29.13.11 for
, .
.
-
29.13.12 for
, .
.
-
29.13.13 for
, .
.
-
29.13.14 for
, .
.
-
29.13.15 for
, .
.
-
29.13.16 for
, .
.
-
29.13.17 for
, .
.
-
29.13.18 for
, .
.
-
29.13.19 for
, .
.
-
29.13.20 for
, .
.
-
29.13.21
for ,
. ,
.
-
29.13.22
for ,
. .
-
29.13.23
for ,
. ,
.
-
30 Spheroidal Wave Functions
-
30.7.1 ,
, .
-
30.7.2 ,
, .
-
30.7.3 ,
, .
-
30.7.4 ,
, .
-
30.7.5 , , .
-
30.7.6 , , .
-
30.7.7 , , .
-
30.7.8 , , .
-
30.7.9 ,
, .
-
30.7.10 ,
, .
-
30.7.11 , , .
-
30.7.12 , , .
-
30.7.13 , for ,
.
-
30.7.14 , , .
-
30.7.15 .
-
30.7.16 ,
, .
-
30.7.17 ,
, .
-
30.7.18 ,
, .
-
30.7.19 ,
, .
-
30.7.20 ,
, .
-
30.7.21 ,
, .
-
30.11.1 ,
, .
-
30.11.2 ,
, .
-
30.11.3 ,
, .
-
30.11.4 ,
, .
-
32 Painlevé Transcendents
-
32.3.1 , , , , , .
-
32.3.2 , , , , , , .
-
32.3.3 , , , .
-
32.3.4 , , , .
-
32.3.5 , , , .
-
32.3.6 , , , .
-
32.3.7 , , , .
-
32.3.8 , , .
-
32.3.9 , , , .
-
32.3.10 , , , .
-
33 Coulomb Functions
-
33.3.1 , ,
, .
-
33.3.2 , ,
, .
-
33.3.3 ,
,
, .
-
33.3.4 ,
,
, .
-
33.3.5 , ,
,
, .
-
33.3.6 , ,
,
, .
-
33.3.7 ,
, .
-
33.3.8 ,
, .
-
33.15.1 ,
.
-
33.15.2 ,
.
-
33.15.3 ,
.
-
33.15.4 ,
.
-
33.15.5 ,
.
-
33.15.6 , .
-
33.15.7 , .
-
33.15.8 , .
-
33.15.9 , .
-
33.15.10 , .
-
33.15.11 , .
-
34 3j, 6j, 9j Symbols
-
36 Integrals with Coalescing Saddles
-
(a) Density plot.
(b) 3D plot.
36.3.1 Modulus of Pearcey integral .
-
(a) Density plot.
(b) 3D plot.
36.3.2 Modulus of swallowtail canonical integral function
.
-
(a) Density plot.
(b) 3D plot.
36.3.3 Modulus of swallowtail canonical integral function
.
-
(a) Density plot.
(b) 3D plot.
36.3.4 Modulus of swallowtail canonical integral function
.
-
(a) Density plot.
(b) 3D plot.
36.3.5 Modulus of swallowtail canonical integral function
.
-
(a) Density plot.
(b) 3D plot.
36.3.6 Modulus of elliptic umbilic canonical integral function
.
-
(a) Density plot.
(b) 3D plot.
36.3.7 Modulus of elliptic umbilic canonical integral function
.
-
(a) Density plot.
(b) 3D plot.
36.3.8 Modulus of elliptic umbilic canonical integral function
.
-
(a) Density plot.
(b) 3D plot.
36.3.9 Modulus of hyperbolic umbilic canonical integral function
.
-
(a) Density plot.
(b) 3D plot.
36.3.10 Modulus of hyperbolic umbilic canonical integral function
.
-
(a) Density plot.
(b) 3D plot.
36.3.11 Modulus of hyperbolic umbilic canonical integral function
.
-
(a) Density plot.
(b) 3D plot.
36.3.12 Modulus of hyperbolic umbilic canonical integral function
.
-
(a) Contour plot, at intervals of .
(b) Density plot.
36.3.13 Phase of Pearcey integral .
-
(a) .
(b) .
(c) .
(d) .
36.3.14 Density plots of phase of swallowtail canonical integrals.
-
(a) Contour plot.
(b) Density plot.
36.3.15 Phase of elliptic umbilic canonical integral
.
-
(a) Contour plot.
(b) Density plot.
36.3.16 Phase of elliptic umbilic canonical integral
.
-
(a) Contour plot.
(b) Density plot.
36.3.17 Phase of elliptic umbilic canonical integral
.
-
(a) Contour plot.
(b) Density plot.
36.3.18 Phase of hyperbolic umbilic canonical integral
.
-
(a) Contour plot.
(b) Density plot.
36.3.19 Phase of hyperbolic umbilic canonical integral
.
-
(a) Contour plot.
(b) Density plot.
36.3.20 Phase of hyperbolic umbilic canonical integral
.
-
(a) Contour plot.
(b) Density plot.
36.3.21 Phase of hyperbolic umbilic canonical integral
.
-
36.4.1 Bifurcation set of cusp catastrophe.
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36.4.2 Bifurcation set of swallowtail catastrophe.
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36.4.3 Bifurcation set of elliptic umbilic catastrophe.
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36.4.4 Bifurcation set of hyperbolic umbilic catastrophe.
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36.5.1 Cusp catastrophe.
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36.5.2 Swallowtail catastrophe with .
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36.5.3 Swallowtail catastrophe with .
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36.5.4 Swallowtail catastrophe with .
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36.5.5 Elliptic
umbilic catastrophe with .
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36.5.6 Hyperbolic
umbilic catastrophe with .
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36.5.7 Sheets of the Stokes surface for the swallowtail catastrophe
(colored and with mesh) and the bifurcation set (gray).
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36.5.8 Sheets of the Stokes surface for the elliptic umbilic catastrophe.
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36.5.9 Sheets of the Stokes surface for the hyperbolic umbilic catastrophe
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36.13.1 Kelvin’s ship wave pattern.