relation to elliptic integrals

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11: 23.6 Relations to Other Functions

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§23.6(iv) Elliptic Integrals

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

12: Bille C. Carlson

… ►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. …

13: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

… ►The following two multivariate hypergeometric series apply to each of the integrals (19.16.14)–(19.16.18) and (19.16.20)–(19.16.23): … ►Then $T_N$ has at most one term if $N\le 5$ in the series for $R_F$. For $R_J$ and $R_D$, $T_N$ has at most one term if $N\le 3$, and two terms if $N=4$ or 5. … ►

14: 19.35 Other Applications

… ►Generalizations of elliptic integrals appear in analysis of modular theorems of Ramanujan (Anderson et al. (2000)); analysis of Selberg integrals (Van Diejen and Spiridonov (2001)); use of Legendre’s relation (19.7.1) to compute $\pi$ to high precision (Borwein and Borwein (1987, p. 26)). …

15: 15.9 Relations to Other Functions

§15.9 Relations to Other Functions

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§15.9(i) Orthogonal Polynomials

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Jacobi

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Meixner

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§15.9(v) Complete Elliptic Integrals

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16: 19.36 Methods of Computation

§19.36 Methods of Computation

… ►Because of cancellations in (19.26.21) it is advisable to compute $R_G$ from $R_F$ and $R_D$ by (19.21.10) or else to use §19.36(ii). ►Legendre’s integrals can be computed from symmetric integrals by using the relations in §19.25(i). … ►Descending Gauss transformations of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ (see (19.8.20)) are used in Fettis (1965) to compute a large table (see §19.37(iii)). … ►

17: Errata

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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 $F(a,a;a+1;\frac{1}{2})$, (15.4.34), (19.5.4_1), (19.5.4_2) and (19.5.4_3).

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18: 19.21 Connection Formulas

§19.21 Connection Formulas

… ►Legendre’s relation (19.7.1) can be written … ►If and $y=z+1$, 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 $p$ of $R_J$ from either circular region to the other, or from either hyperbolic region to the other (§19.20(iii)). …

19: 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 , … ►A related hyperbolic series is …where $E^{\prime }=E^{\prime }\left(k\right)$ is defined by §19.2.9. …

20: 19.7 Connection Formulas

§19.7 Connection Formulas

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Reciprocal-Modulus Transformation

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Imaginary-Modulus Transformation

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§19.7(iii) Change of Parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$

►There are three relations connecting $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ and $\mathrm{\Pi }(\varphi ,{\omega }^2,k)$, where ${\omega }^2$ is a rational function of ${\alpha }^2$. …