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 by … ►The number of terms in can be greatly reduced by using variables with chosen to make . … ►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 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 from and 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 , then as (19.21.6) reduces to Legendre’s relation (19.21.1). … ► is symmetric only in and , but either (nonzero) or (nonzero) can be moved to the third position by using … ►§19.21(iii) Change of Parameter of
… ►Change-of-parameter relations can be used to shift the parameter of 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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