symmetric elliptic%0Aintegrals

SearchAdvancedHelp

(0.004 seconds)

11—20 of 738 matching pages

11: 19.1 Special Notation

… ►All derivatives are denoted by differentials, not by primes. … ►We use also the function $D(\varphi ,k)$, introduced by Jahnke et al. (1966, p. 43). … ►However, it should be noted that in Chapter 8 of Abramowitz and Stegun (1964) the notation used for elliptic integrals differs from Chapter 17 and is consistent with that used in the present chapter and the rest of the NIST Handbook and DLMF. … ► $R_F(x,y,z)$, $R_G(x,y,z)$, and $R_J(x,y,z,p)$ are the symmetric (in $x$, $y$, and $z$) integrals of the first, second, and third kinds; they are complete if exactly one of $x$, $y$, and $z$ is identically 0. ► $R_{-a}(b_1,b_2,\mathrm{\dots },b_n;z_1,z_2,\mathrm{\dots },z_n)$ is a multivariate hypergeometric function that includes all the functions in (19.1.3). …

12: 19.25 Relations to Other Functions

… ►

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

… ►

§19.25(v) Jacobian Elliptic Functions

… ► ►

§19.25(vi) Weierstrass Elliptic Functions

… ►

13: 22.14 Integrals

… ►Lastly, with , … ►For alternative, and symmetric, formulations of the results in this subsection see Carlson (2006a). … ►For indefinite integrals of squares and products of even powers of Jacobian functions in terms of symmetric elliptic integrals, see Carlson (2006b). … ►In (22.14.13)–(22.14.15), . … …

14: 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 …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 $R_J$

…

15: 19.18 Derivatives and Differential Equations

… ►

§19.18(i) Derivatives

… ►

§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: …

16: 19.37 Tables

§19.37 Tables

… ►($F(\varphi ,k)$ is presented as $\mathrm{\Pi }(\varphi ,0,k)$.) … ► ►

§19.37(iv) Symmetric Integrals

… ►For check values of symmetric integrals with real or complex variables to 14S see Carlson (1995).

17: 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). When the differences are moderately small, the iteration is stopped, the elementary symmetric functions of certain differences are calculated, and a polynomial consisting of a fixed number of terms of the sum in (19.19.7) is evaluated. …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). …

18: 19.24 Inequalities

… ►

§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. …

19: 19.22 Quadratic Transformations

… ►

Bartky’s Transformation

… ►

§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). … ► … ►

20: 19.27 Asymptotic Approximations and Expansions

§19.27 Asymptotic Approximations and Expansions

… ►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 $p>0$. … ►Although they are obtained (with some exceptions) by approximating uniformly the integrand of each elliptic integral, some occur also as the leading terms of known asymptotic series with error bounds (Wong (1983, §4), Carlson and Gustafson (1985), López (2000, 2001)). … ►