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11: 19.36 Methods of Computation

§19.36 Methods of Computation

… ►Numerical differences between the variables of a symmetric integral can be reduced in magnitude by successive factors of 4 by repeated applications of the duplication theorem, as shown by (19.26.18). …where the elementary symmetric functions $E_s$ are defined by (19.19.4). … ►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. …

12: 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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13: 19.21 Connection Formulas

§19.21 Connection Formulas

… ► $R_D(x,y,z)$ is symmetric only in $x$ and $y$, but either (nonzero) $x$ or (nonzero) $y$ can be moved to the third position by using …Because $R_G$ is completely symmetric, $x,y,z$ can be permuted on the right-hand side of (19.21.10) so that $(x-z)(y-z)\le 0$ if the variables are real, thereby avoiding cancellations when $R_G$ is calculated from $R_F$ and $R_D$ (see §19.36(i)). … ►

§19.21(iii) Change of Parameter of $R_J$

… ►If $x=0$, then $\xi =\eta =\mathrm{\infty }$ and $R_C(\xi ,\eta )=0$; hence …

14: 35.2 Laplace Transform

… ►For any complex symmetric matrix $𝐙$, … ►Suppose there exists a constant ${𝐗}_0\in 𝛀$ such that for all $𝐗\in 𝛀$. Then (35.2.1) converges absolutely on the region $\mathrm{\Re }\left(𝐙\right)>{𝐗}_0$, and $g(𝐙)$ is a complex analytic function of all elements $z_{j,k}$ of $𝐙$. … ►Assume that ${\int }_{𝓢}\left|g(𝐔+\mathrm{i}𝐕)\right|d𝐕$ converges, and also that its limit as $𝐔\to \mathrm{\infty }$ is $0$. …where the integral is taken over all $𝐙=𝐔+\mathrm{i}𝐕$ such that $𝐔>{𝐗}_0$ and $𝐕$ ranges over $𝓢$. …

15: 19.19 Taylor and Related Series

§19.19 Taylor and Related Series

►For $N=0,1,2,\mathrm{\dots }$ define the homogeneous hypergeometric polynomial … ►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 $E_1(𝐙)=0$. … ►

16: 19.24 Inequalities

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§19.24(i) Complete Integrals

►The condition $y\le z$ for (19.24.1) and (19.24.2) serves only to identify $y$ as the smaller of the two nonzero variables of a symmetric function; it does not restrict validity. … ► ►

§19.24(ii) Incomplete Integrals

… ►The same reference also gives upper and lower bounds for symmetric integrals in terms of their elementary degenerate cases. …

17: 19.22 Quadratic Transformations

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Bartky’s Transformation

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§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)

►The AGM, $M(a_0,g_0)$, of two positive numbers $a_0$ and $g_0$ is defined in §19.8(i). … ► … ►

18: 19.38 Approximations

§19.38 Approximations

►Minimax polynomial approximations (§3.11(i)) for $K\left(k\right)$ and $E\left(k\right)$ in terms of $m=k^2$ with can be found in Abramowitz and Stegun (1964, §17.3) with maximum absolute errors ranging from 4×10⁻⁵ to 2×10⁻⁸. Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. … ►Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …

19: 19.27 Asymptotic Approximations and Expansions

§19.27 Asymptotic Approximations and Expansions

… ►Assume $x$, $y$, and $z$ are real and nonnegative and at most one of them is 0. … ►Assume $x$ and $y$ are real and nonnegative, at most one of them is 0, and $z>0$. … ►Assume $x$, $y$, and $z$ are real and nonnegative, at most one of them is 0, and $p>0$. … ►

20: 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: …