relation to elliptic integrals

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21: 36.5 Stokes Sets

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§36.5(ii) Cuspoids

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§36.5(iii) Umbilics

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Elliptic Umbilic Stokes Set (Codimension three)

►This consists of three separate cusp-edged sheets connected to the cusp-edged sheets of the bifurcation set, and related by rotation about the $z$-axis by $2\pi /3$. … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …

22: Bibliography G

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Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

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External Links:

MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

Encodings:

BibTeX

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Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

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External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

Encodings:

BibTeX

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Definite integrals of the complete elliptic integral $K$ . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

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Automatic reduction of elliptic integrals using Carlson’s relations. Math. Comp. 71 (237), pp. 311–318.

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External Links:

ISSN 0025-5718, Document, MathReview (David A. Richter), ZentralBlatt

Referenced by:

§19.29(ii).

Encodings:

BibTeX

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Fourier transforms related to a root system of rank 1. Transform. Groups 12 (1), pp. 77–116.

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External Links:

ISSN 1083-4362, Link, MathReview (Genkai Zhang), ZentralBlatt

Referenced by:

§18.28(xi), §18.38(iii).

Encodings:

BibTeX

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23: 20.11 Generalizations and Analogs

… ►As in §20.11(ii), the modulus $k$ of elliptic integrals (§19.2(ii)), Jacobian elliptic functions (§22.2), and Weierstrass elliptic functions (§23.6(ii)) can be expanded in $q$-series via (20.9.1). However, in this case $q$ is no longer regarded as an independent complex variable within the unit circle, because $k$ is related to the variable $\tau =\tau (k)$ of the theta functions via (20.9.2). … ►For applications to rapidly convergent expansions for $\pi$ see Chudnovsky and Chudnovsky (1988), and for applications in the construction of elliptic-hypergeometric series see Rosengren (2004). … ►For specialization to the one-dimensional theta functions treated in the present chapter, see Rauch and Lebowitz (1973) and §21.7(iii). … ►Such sets of twelve equations include derivatives, differential equations, bisection relations, duplication relations, addition formulas (including new ones for theta functions), and pseudo-addition formulas. …

24: 29.2 Differential Equations

… ►For $\mathrm{sn}(z,k)$ see §22.2. This equation has regular singularities at the points $2pK+(2q+1)\mathrm{i}{K}^{\prime }$, where $p,q\in \mathbb{Z}$, and $K$, $K^{\prime }$ are the complete elliptic integrals of the first kind with moduli $k$, $k^{\prime }(={(1-k^2)}^{1/2})$, respectively; see §19.2(ii). … ►

§29.2(ii) Other Forms

… ►we have …For the Weierstrass function $\mathrm{\wp }$ see §23.2(ii). …

25: 19.1 Special Notation

… ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). …

26: 28.32 Mathematical Applications

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§28.32(i) Elliptical Coordinates and an Integral Relationship

►If the boundary conditions in a physical problem relate to the perimeter of an ellipse, then elliptical coordinates are convenient. … ►This leads to integral equations and an integral relation between the solutions of Mathieu’s equation (setting $\zeta =\mathrm{i}\xi$, $z=\eta$ in (28.32.3)). … ►defines a solution of Mathieu’s equation, provided that (in the case of an improper curve) the integral converges with respect to $z$ uniformly on compact subsets of $ℂ$. … ► …

27: 19.14 Reduction of General Elliptic Integrals

§19.14 Reduction of General Elliptic Integrals

… ►Legendre (1825–1832) showed that every elliptic integral can be expressed in terms of the three integrals in (19.1.2) supplemented by algebraic, logarithmic, and trigonometric functions. The classical method of reducing (19.2.3) to Legendre’s integrals is described in many places, especially Erdélyi et al. (1953b, §13.5), Abramowitz and Stegun (1964, Chapter 17), and Labahn and Mutrie (1997, §3). …A similar remark applies to the transformations given in Erdélyi et al. (1953b, §13.5) and to the choice among explicit reductions in the extensive table of Byrd and Friedman (1971), in which one limit of integration is assumed to be a branch point of the integrand at which the integral converges. …

28: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

… ►is called an elliptic integral. … ►

§19.2(ii) Legendre’s Integrals

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§19.2(iii) Bulirsch’s Integrals

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§19.2(iv) A Related Function: $R_C(x,y)$

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29: 22.20 Methods of Computation

… ►This formula for $\mathrm{dn}$ becomes unstable near $x=K$. … ►To compute $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$ to 10D when $x=0.8$, $k=0.65$. … ►

§22.20(vi) Related Functions

… ►Alternatively, Sala (1989) shows how to apply the arithmetic-geometric mean to compute $\mathrm{am}(x,k)$. ►Jacobi’s epsilon function can be computed from its representation (22.16.30) in terms of theta functions and complete elliptic integrals; compare §20.14. …

30: 19.18 Derivatives and Differential Equations

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§19.18(i) Derivatives

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§19.18(ii) Differential Equations

… ►and two similar equations obtained by permuting $x,y,z$ in (19.18.10). ►More concisely, if $v=R_{-a}(𝐛;𝐳)$, then each of (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23) satisfies Euler’s homogeneity relation: … ►The next four differential equations apply to the complete case of $R_F$ and $R_G$ in the form $R_{-a}(\frac{1}{2},\frac{1}{2};z_1,z_2)$ (see (19.16.20) and (19.16.23)). …