relation to Legendre elliptic integrals

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11: 22.11 Fourier and Hyperbolic Series

§22.11 Fourier and Hyperbolic Series

… ►Next, with

E=E\left(k\right)

denoting the complete elliptic integral of the second kind (§19.2(ii)) and

q\exp\left(2|\Im\zeta|\right)<1

, … ►A related hyperbolic series is …where

{E^{\prime}}={E^{\prime}}\left(k\right)

is defined by §19.2.9. …

12: 19.21 Connection Formulas

§19.21 Connection Formulas

… ►Legendre’s relation (19.7.1) can be written … ►If

0 <mrow> <mn>0</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>z</mi> </mrow> </math> and <math class=

, then as

p\to 0

(19.21.6) reduces to Legendre’s relation (19.21.1). … ►

§19.21(iii) Change of Parameter of $R_{J}$

… ►Change-of-parameter relations can be used to shift the parameter

p

R_{J}

from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). …

… ►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. Symmetric integrals and their degenerate cases allow greatly shortened integral tables and improved algorithms for numerical computation. … ►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. … ►

19 Elliptic Integrals

…

14: 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). …The choice among 21 transformations for final reduction to Legendre’s normal form depends on inequalities involving the limits of integration and the zeros of the cubic or quartic polynomial. …

15: 20.9 Relations to Other Functions

§20.9 Relations to Other Functions

►

§20.9(i) Elliptic Integrals

… ►

§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. ►The relations (20.9.1) and (20.9.2) between

k

and

\tau

(or

q

) are solutions of Jacobi’s inversion problem; see Baker (1995) and Whittaker and Watson (1927, pp. 480–485). …

16: 22.2 Definitions

§22.2 Definitions

… ►where

K\left(k\right)

{K^{\prime}}\left(k\right)

are defined in §19.2(ii). … ► … ►The Jacobian functions are related in the following way. …

\operatorname{ss}\left(z,k\right)=1

. …

17: 23.15 Definitions

§23.15 Definitions

… ►(Some references refer to

2\ell

as the level). …If, in addition,

f(\tau)\to 0

q\to 0

, then

f(\tau)

is called a cusp form. … ►

Elliptic Modular Function

… ►

18: 29.2 Differential Equations

… ►For

\operatorname{sn}\left(z,k\right)

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

\wp

see §23.2(ii). …

19: Errata

… ►

Subsection 19.2(ii) and Equation (19.2.9)

The material surrounding (19.2.8), (19.2.9) has been updated so that the complementary complete elliptic integrals of the first and second kind are defined with consistent multivalued properties and correct analytic continuation. In particular, (19.2.9) has been corrected to read

19.2.9

{K^{\prime}}\left(k\right)=\begin{cases}K\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ K\left(k^{\prime}\right)\mp 2\mathrm{i}K\left(-k\right),&\tfrac{1}{2}\pi<\pm% \operatorname{ph}k<\pi,\end{cases}

{E^{\prime}}\left(k\right)=\begin{cases}E\left(k^{\prime}\right),&|% \operatorname{ph}k|\leq\tfrac{1}{2}\pi,\\ E\left(k^{\prime}\right)\mp 2\mathrm{i}(K\left(-k\right)-E\left(-k\right)),&% \tfrac{1}{2}\pi<\pm\operatorname{ph}k<\pi\end{cases}

… ►

Chapters 1 Algebraic and Analytic Methods, 10 Bessel Functions, 14 Legendre and Related Functions, 18 Orthogonal Polynomials, 29 Lamé Functions

Over the preceding two months, the subscript parameters of the Ferrers and Legendre functions, $\mathsf{P}_{n},\mathsf{Q}_{n},P_{n},Q_{n},\boldsymbol{Q}_{n}$ and the Laguerre polynomial, $L_{n}$ , were incorrectly displayed as superscripts. Reported by Roy Hughes on 2022-05-23

… ►

Additions

Section: 15.9(v) Complete Elliptic Integrals. Equations: (11.11.9_5), (11.11.13_5), Intermediate equality in (15.4.27) which relates to $F\left(a,a;a+1;\tfrac{1}{2}\right)$ , (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

… ►

Chapters 14 Legendre and Related Functions, 15 Hypergeometric Function

The Gegenbauer function $C^{(\lambda)}_{\alpha}\left(z\right)$ , was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C^{(\lambda)}_{n}\left(z\right)$ . In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

… ►

Equation (14.2.7)

The Wronskian was generalized to include both associated Legendre and Ferrers functions.

…

20: 23.6 Relations to Other Functions

… ►

§23.6(ii) Jacobian Elliptic Functions

… ►

§23.6(iii) General Elliptic Functions

… ►

§23.6(iv) Elliptic Integrals

… ►For relations to symmetric elliptic integrals see §19.25(vi). … ►