List of Figures
- 1 Algebraic and Analytic Methods
- 2 Asymptotic Approximations
- 3 Numerical Methods
-
4 Elementary Functions
- 4.2.1 Branch cut for and .
- 4.3.1 and .
- 4.3.2 Conformal mapping of exponential and logarithm.
- 4.3.3 .
- 4.3.4 .
- 4.13.1 Branches , of the Lambert -function.
- 4.15.1 and .
- 4.15.2 and .
- 4.15.3 and .
- 4.15.4 and .
- 4.15.5 and .
- 4.15.6 and .
- 4.15.7 Conformal mapping of sine and inverse sine.
- 4.15.8 .
- 4.15.9 .
- 4.15.10 .
- 4.15.11 .
- 4.15.12 .
- 4.15.13 .
- 4.16.1 Quadrants for the angle .
- 4.23.1 Branch cuts for the inverse trigonometric functions.
- 4.29.1 and .
- 4.29.2 and .
- 4.29.3 and .
- 4.29.4 and .
- 4.29.5 and .
- 4.29.6 and .
- 4.37.1 Branch cuts for the inverse hyperbolic functions.
- 4.42.1 Planar right triangle.
- 4.42.2 Planar triangle.
- 4.42.3 Spherical triangle.
- 5 Gamma Function
- 6 Exponential, Logarithmic, Sine, and Cosine Integrals
- 7 Error Functions, Dawson’s and Fresnel Integrals
-
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
-
10 Bessel Functions
- 10.3.1 , , , , .
- 10.3.2 , , , .
- 10.3.3 , , , .
- 10.3.4 , , .
- 10.3.17 , , .
- 10.3.18 , , .
- 10.3.19 , , .
- 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 , , .
- 10.20.1 -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 , , , , .
- 10.26.7 , , .
- 10.26.8 , , .
- 10.26.9 , , .
- 10.26.10 , .
- 10.26.3 , , .
- 10.26.4 , , .
- 10.26.5 , , .
- 10.26.6 , , .
- 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 , , .
- 11.3.13 , , .
- 11.3.14 , , .
- 11.3.15 , , .
- 11.3.16 , , .
- 11.3.5 , , .
- 11.3.6 , , .
- 11.3.7 , , .
- 11.3.8 , , .
- 11.3.9 , , .
- 11.3.10 , , .
- 11.3.11 , , .
- 11.3.12 , , .
- 11.3.17 , , .
- 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
- 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 , .
- 14.4.17 , .
- 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 , .
- 14.4.13 , .
- 14.4.14 , .
- 14.4.15 , .
- 14.4.16 , .
- 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
- 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
-
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
-
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 , , , , , .
- 22.3.22 .
- 22.3.23 .
- 22.3.26 Density plot of .
- 22.3.27 Density plot of .
- 22.3.28 Density plot of .
- 22.3.29 Density plot of .
- 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 , , .
- 22.3.24 .
- 22.3.25 .
- 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
- 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 , .
- 29.4.13 , , .
- 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 , , .
- 29.4.9 .
- 29.4.10 .
- 29.4.11 .
- 29.4.12 .
- 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 , , .
- 30.7.11 , , .
- 30.7.12 , , .
- 30.7.13 , for , .
- 30.7.14 , , .
- 30.7.9 , , .
- 30.7.10 , , .
- 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
- 33 Coulomb Functions
- 34 3j, 6j, 9j Symbols
-
36 Integrals with Coalescing Saddles
- 36.3.1 Modulus of Pearcey integral .
- 36.3.2 Modulus of swallowtail canonical integral function .
- 36.3.3 Modulus of swallowtail canonical integral function .
- 36.3.4 Modulus of swallowtail canonical integral function .
- 36.3.5 Modulus of swallowtail canonical integral function .
- 36.3.6 Modulus of elliptic umbilic canonical integral function .
- 36.3.7 Modulus of elliptic umbilic canonical integral function .
- 36.3.8 Modulus of elliptic umbilic canonical integral function .
- 36.3.9 Modulus of hyperbolic umbilic canonical integral function .
- 36.3.10 Modulus of hyperbolic umbilic canonical integral function .
- 36.3.11 Modulus of hyperbolic umbilic canonical integral function .
- 36.3.12 Modulus of hyperbolic umbilic canonical integral function .
- 36.3.13 Phase of Pearcey integral .
- 36.3.14 Density plots of phase of swallowtail canonical integrals.
- 36.3.15 Phase of elliptic umbilic canonical integral .
- 36.3.16 Phase of elliptic umbilic canonical integral .
- 36.3.17 Phase of elliptic umbilic canonical integral .
- 36.3.18 Phase of hyperbolic umbilic canonical integral .
- 36.3.19 Phase of hyperbolic umbilic canonical integral .
- 36.3.20 Phase of hyperbolic umbilic canonical integral .
- 36.3.21 Phase of hyperbolic umbilic canonical integral .
- 36.4.1 Bifurcation set of cusp catastrophe.
- 36.4.2 Bifurcation set of swallowtail catastrophe.
- 36.4.3 Bifurcation set of elliptic umbilic catastrophe.
- 36.4.4 Bifurcation set of hyperbolic umbilic catastrophe.
- 36.5.1 Cusp catastrophe.
- 36.5.2 Swallowtail catastrophe with .
- 36.5.3 Swallowtail catastrophe with .
- 36.5.4 Swallowtail catastrophe with .
- 36.5.5 Elliptic umbilic catastrophe with .
- 36.5.6 Hyperbolic umbilic catastrophe with .
- 36.5.7 Sheets of the Stokes surface for the swallowtail catastrophe (colored and with mesh) and the bifurcation set (gray).
- 36.5.8 Sheets of the Stokes surface for the elliptic umbilic catastrophe.
- 36.5.9 Sheets of the Stokes surface for the hyperbolic umbilic catastrophe
- 36.13.1 Kelvin’s ship wave pattern.