List of Tables 1 Algebraic and Analytic Methods 1.14.1 Fourier transforms. 1.14.2 Fourier cosine transforms. 1.14.3 Fourier sine transforms. 1.14.4 Laplace transforms. 1.14.5 Mellin transforms. 2 Asymptotic Approximations 2.5.1 Domains of convergence for Mellin transforms. 2.5.2 Domains of analyticity for Mellin transforms. 2.11.1 Whittaker functions with Levin’s transformation. 3 Numerical Methods 3.5.1 Nodes and weights for the 5-point Gauss–Legendre formula. 3.5.2 Nodes and weights for the 10-point Gauss–Legendre formula. 3.5.3 Nodes and weights for the 20-point Gauss–Legendre formula. 3.5.4 Nodes and weights for the 40-point Gauss–Legendre formula. 3.5.5 Nodes and weights for the 80-point Gauss–Legendre formula. 3.5.6 Nodes and weights for the 5-point Gauss–Laguerre formula. 3.5.7 Nodes and weights for the 10-point Gauss–Laguerre formula. 3.5.8 Nodes and weights for the 15-point Gauss–Laguerre formula. 3.5.9 Nodes and weights for the 20-point Gauss–Laguerre formula. 3.5.10 Nodes and weights for the 5-point Gauss–Hermite formula. 3.5.11 Nodes and weights for the 10-point Gauss–Hermite formula. 3.5.12 Nodes and weights for the 15-point Gauss–Hermite formula. 3.5.13 Nodes and weights for the 20-point Gauss–Hermite formula. 3.5.14 Nodes and weights for the 5-point Gauss formula for the logarithmic weight function. 3.5.15 Nodes and weights for the 10-point Gauss formula for the logarithmic weight function. 3.5.16 Nodes and weights for the 15-point Gauss formula for the logarithmic weight function. 3.5.17 Nodes and weights for the 20-point Gauss formula for the logarithmic weight function. 3.5.17_5 Recurrence coefficients in (3.5.30) and (3.5.30_5) for monic versions pn(x) and orthonormal versions qn(x) of the classical orthogonal polynomials. 3.5.18 Nodes and weights for the 5-point complex Gauss quadrature formula with s=1. 3.5.19 Laplace transform inversion. 3.5.20 Composite trapezoidal rule for the integral (3.5.45) with λ=10. 3.5.21 Cubature formulas for disk and square. 3.6.1 Weber function wn=𝐄n(1) computed by Olver’s algorithm. 3.8.1 Newton’s rule for x−tanx=0. 3.8.2 Newton’s rule for z4−1=0. 3.8.3 Bairstow’s method for factoring z4−2z2+1. 3.9.1 Shanks’ transformation for sn=∑j=1n(−1)j+1j−2. 3.10.1 Quotient-difference scheme. 3.11.1 Coefficients pj, qj for the minimax rational approximation R3,3(x). 4 Elementary Functions 4.16.1 Signs of the trigonometric functions in the four quadrants. 4.16.2 Trigonometric functions: quarter periods and change of sign. 4.16.3 Trigonometric functions: interrelations. 4.17.1 Trigonometric functions: values at multiples of 112π. 4.23.1 Inverse trigonometric functions. 4.30.1 Hyperbolic functions: interrelations. 4.31.1 Hyperbolic functions: values at multiples of 12πi. 5 Gamma Function 5.4.1 Γ′(xn)=ψ(xn)=0. 7 Error Functions, Dawson’s and Fresnel Integrals 7.13.1 Zeros xn+iyn of erfz. 7.13.2 Zeros xn+iyn of erfcz. 7.13.3 Complex zeros xn+iyn of C(z). 7.13.4 Complex zeros xn+iyn of S(z). 8 Incomplete Gamma and Related Functions 8.13.1 Double zeros (an∗,xn∗) of γ∗(a,x). 9 Airy and Related Functions 9.2.1 Numerically satisfactory pairs of solutions of Airy’s equation. 9.7.1 χ(n). 9.9.1 Zeros of Ai and Ai′. 9.9.2 Real zeros of Bi and Bi′. 9.9.3 Complex zeros of Bi. 9.9.4 Complex zeros of Bi′. 10 Bessel Functions 10.2.1 Numerically satisfactory pairs of solutions of Bessel’s equation. 10.25.1 Numerically satisfactory pairs of solutions of the modified Bessel’s equation. 15 Hypergeometric Function 15.8.1 Quadratic transformations of the hypergeometric function. 18 Orthogonal Polynomials 18.3.1 Orthogonality properties for classical OP’s. 18.5.1 Classical OP’s: Rodrigues formulas (18.5.5). 18.6.1 Classical OP’s: symmetry and special values. 18.8.1 Classical OP’s: differential equations A(x)f′′(x)+B(x)f′(x)+C(x)f(x)+λnf(x)=0. 18.9.1 Classical OP’s: recurrence relations (18.9.1). 18.9.2 Classical OP’s: recurrence relations (18.9.2_1). 18.10.1 Classical OP’s: contour integral representations (18.10.8). 18.19.1 Orthogonality properties for Hahn, Krawtchouk, Meixner, and Charlier OP’s. 18.19.2 Hahn, Krawtchouk, Meixner, and Charlier OP’s: leading coefficients. 18.20.1 Krawtchouk, Meixner, and Charlier OP’s: Rodrigues formulas (18.20.1). 18.22.1 Recurrence relations (18.22.2) for Krawtchouk, Meixner, and Charlier polynomials. 18.22.2 Difference equations (18.22.12) for Krawtchouk, Meixner, and Charlier polynomials. 18.25.1 Wilson class OP’s: transformations of variable, orthogonality ranges, and parameter constraints. 18.25.2 Wilson class OP’s: leading coefficients. 18.39.1 Typical Non-Classical Weight Functions Of Use In DVR Applicationsa 22 Jacobian Elliptic Functions 22.4.1 Periods and poles of Jacobian elliptic functions. 22.4.2 Periods and zeros of Jacobian elliptic functions. 22.4.3 Half- or quarter-period shifts of variable for the Jacobian elliptic functions. 22.5.1 Jacobian elliptic function values 22.5.2 Other special values of Jacobian elliptic functions. 22.5.3 Limiting forms of Jacobian elliptic functions as k→0. 22.5.4 Limiting forms of Jacobian elliptic functions as k→1. 22.6.1 Jacobi’s imaginary transformation of Jacobian elliptic functions. 22.13.1 Derivatives of Jacobian elliptic functions with respect to variable. 24 Bernoulli and Euler Polynomials 24.2.1 Bernoulli and Euler numbers. 24.2.2 Bernoulli and Euler polynomials. 24.2.3 Bernoulli numbers Bn=N/D. 24.2.4 Euler numbers En. 24.2.5 Coefficients bn,k of the Bernoulli polynomials. 24.2.6 Coefficients en,k of the Euler polynomials. 24.15.1 Genocchi and Tangent numbers. 26 Combinatorial Analysis 26.2.1 Partitions p(n). 26.3.1 Binomial coefficients (mn). 26.3.2 Binomial coefficients (m+nm) for lattice paths. 26.4.1 Multinomials and partitions. 26.5.1 Catalan numbers. 26.6.1 Delannoy numbers D(m,n). 26.6.2 Motzkin numbers M(n). 26.6.3 Narayana numbers N(n,k). 26.6.4 Schröder numbers r(n). 26.7.1 Bell numbers. 26.8.1 Stirling numbers of the first kind s(n,k). 26.8.2 Stirling numbers of the second kind S(n,k). 26.9.1 Partitions pk(n). 26.10.1 Partitions restricted by difference conditions. 26.12.1 Plane partitions. 26.14.1 Eulerian numbers ⟨nk⟩. 26.17.1 The twelvefold way. 27 Functions of Number Theory 27.2.1 Primes. 27.2.2 Functions related to division. 28 Mathieu Functions and Hill’s Equation 28.1.1 Notations for parameters in Mathieu’s equation. 28.2.1 Eigenvalues of Mathieu’s equation. 28.2.2 Eigenfunctions of Mathieu’s equation. 28.6.1 Radii of convergence for power-series expansions of eigenvalues of Mathieu’s equation. 29 Lamé Functions 29.3.1 Eigenvalues of Lamé’s equation. 29.3.2 Lamé functions. 29.12.1 Lamé polynomials. 36 Integrals with Coalescing Saddles 36.6.1 Special cases of scaling exponents for cuspoids. 36.7.1 Zeros of cusp diffraction catastrophe to 5D.
