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

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§19.25(v) Jacobian Elliptic Functions

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§19.25(vii) Hypergeometric Function

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2: 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. …

3: 23.6 Relations to Other Functions

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

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 $E_s(𝐳)$ by … ►The number of terms in $T_N$ can be greatly reduced by using variables $𝐙=\mathrm{𝟏}-(𝐳/A)$ with $A$ chosen to make $E_1(𝐙)=0$. … ►

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

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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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§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 $2\pi /3$. One of the sheets is symmetrical under reflection in the plane $y=0$, and is given by … ►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. …

10: 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). … ►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)). … ►Similarly, the function $u=R_{-a}(\frac{1}{2},\frac{1}{2};x+\mathrm{i}y,x-\mathrm{i}y)$ satisfies an equation of axially symmetric potential theory: …