symmetric%20elliptic%0Aintegrals

SearchAdvancedHelp

(0.004 seconds)

11—20 of 750 matching pages

11: 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. … ►

12: 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). …

13: 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 $R_J$

…

14: 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

… ►

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: 19.19 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 $E_1(𝐙)=0$. … ►

17: 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. …

18: 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). … ► … ►

19: 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)). … ►

20: 36.5 Stokes Sets

… ►The Stokes set takes different forms for $z=0$, , and $z>0$. ►For $z=0$, the set consists of the two curves … ►

§36.5(iii) Umbilics

►

Elliptic Umbilic Stokes Set (Codimension three)

… ►One of the sheets is symmetrical under reflection in the plane $y=0$, and is given by …