# relation to symmetric elliptic integrals

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##### 1: 19.25 Relations to Other Functions

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###### §19.25(ii) Bulirsch’s Integrals as Symmetric Integrals

… ►###### §19.25(v) Jacobian Elliptic Functions

… ► … ►###### §19.25(vii) Hypergeometric Function

… ►##### 2: Bille C. Carlson

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►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.
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##### 3: 23.6 Relations to Other Functions

##### 4: 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): … ►Define the*elementary symmetric function*${E}_{s}(\mathbf{z})$ by … ►The number of terms in ${T}_{N}$ can be greatly reduced by using variables $\mathbf{Z}=\mathbf{1}-(\mathbf{z}/A)$ with $A$ chosen to make ${E}_{1}(\mathbf{Z})=0$. … ►

##### 5: 19.35 Other Applications

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###### §19.35(i) Mathematical

►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)). ►###### §19.35(ii) Physical

… ►##### 6: 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). … ►Complete cases of Legendre’s integrals and symmetric integrals can be computed with quadratic convergence by the AGM method (including Bartky transformations), using the equations in §19.8(i) and §19.22(ii), respectively. … ►##### 7: 22.15 Inverse Functions

###### §22.15 Inverse Functions

… ►The inverse Jacobian elliptic functions can be defined in an analogous manner to the inverse trigonometric functions (§4.23). … ►###### §22.15(ii) Representations as Elliptic Integrals

… ►For representations of the inverse functions as symmetric elliptic integrals see §19.25(v). …##### 8: 19.21 Connection Formulas

###### §19.21 Connection Formulas

… ►If $$ and $y=z+1$, then as $p\to 0$ (19.21.6) reduces to Legendre’s relation (19.21.1). … ► ${R}_{D}(x,y,z)$ is symmetric only in $x$ and $y$, but either (nonzero) $x$ or (nonzero) $y$ can be moved to the third position by using … ►###### §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)). …##### 9: 36.5 Stokes Sets

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