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31: 20.9 Relations to Other Functions

§20.9 Relations to Other Functions

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§20.9(i) Elliptic Integrals

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§20.9(ii) Elliptic Functions and Modular Functions

►See §§22.2 and 23.6(i) for the relations of Jacobian and Weierstrass elliptic functions to theta functions. … ►

§20.9(iii) Riemann Zeta Function

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32: 18.38 Mathematical Applications

… ►The Askey–Gasper inequality … ► … ►The orthogonality relations (34.5.14) for the $6j$ symbols can be rewritten in terms of orthogonality relations for Racah polynomials as given by (18.25.9)–(18.25.12). … … ►SUSY leads to algebraic simplifications in generating excited states, and partner potentials with closely related energy spectra, from knowledge of a single ground state wave function. …

33: 12.7 Relations to Other Functions

§12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

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§12.7(iii) Modified Bessel Functions

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§12.7(iv) Confluent Hypergeometric Functions

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34: 31.7 Relations to Other Functions

§31.7 Relations to Other Functions

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§31.7(i) Reductions to the Gauss Hypergeometric Function

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§31.7(ii) Relations to Lamé Functions

… ►equation (31.2.1) becomes Lamé’s equation with independent variable $\zeta$; compare (29.2.1) and (31.2.8). …Similar specializations of formulas in §31.3(ii) yield solutions in the neighborhoods of the singularities $\zeta =K$, $K+\mathrm{i}{K}^{\prime }$, and $\mathrm{i}{K}^{\prime }$, where $K$ and $K^{\prime }$ are related to $k$ as in §19.2(ii).

35: Richard B. Paris

… ►His books are Asymptotics of High Order Differential Equations (with A. … ►

8 Incomplete Gamma and Related Functions

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11 Struve and Related Functions

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36: 7.18 Repeated Integrals of the Complementary Error Function

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§7.18(iv) Relations to Other Functions

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Hermite Polynomials

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Confluent Hypergeometric Functions

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Parabolic Cylinder Functions

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Probability Functions

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37: Bille C. Carlson

… ►After the war he returned to Harvard and completed Bachelor’s and Master’s degrees in physics and mathematics. He then went to Oxford as a Rhodes Scholar and completed a doctoral degree in physics. … ►This symmetry led to the development of symmetric elliptic integrals, which are free from the transformations of modulus and amplitude that complicate the Legendre theory. … ►In Symmetry in c, d, n of Jacobian elliptic functions (2004) he found a previously hidden symmetry in relations between Jacobian elliptic functions, which can now take a form that remains valid when the letters c, d, and n are permuted. This invariance usually replaces sets of twelve equations by sets of three equations and applies also to the relation between the first symmetric elliptic integral and the Jacobian functions. …

38: 10.16 Relations to Other Functions

§10.16 Relations to Other Functions

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Elementary Functions

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Parabolic Cylinder Functions

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Confluent Hypergeometric Functions

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Generalized Hypergeometric Functions

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39: 15.19 Methods of Computation

… ►Moreover, it is also possible to accelerate convergence by appropriate choice of $z_0$. … ►

§15.19(ii) Differential Equation

►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. … ►

§15.19(iv) Recurrence Relations

►The relations in §15.5(ii) can be used to compute $F(a,b;c;z)$, provided that care is taken to apply these relations in a stable manner; see §3.6(ii). …

40: 18.34 Bessel Polynomials

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§18.34(i) Definitions and Recurrence Relation

… ►where ${𝗄}_n$ is a modified spherical Bessel function (10.49.9), and … … ►

§18.34(iii) Other Properties

… ►where primes denote derivatives with respect to $x$. …