symmetric elliptic%0Aintegrals

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31: 19.3 Graphics

§19.3 Graphics

… ►See Figures 19.3.1–19.3.6 for complete and incomplete Legendre’s elliptic integrals. … ►In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. … ►

See accompanying text — Figure 19.3.9: $\Re\left(K\left(k\right)\right)$ as a function of complex $k^{2}$ for $-2\leq\Re\left(k^{2}\right)\leq 2$ , $-2\leq\Im\left(k^{2}\right)\leq 2$ . The real part is symmetric under reflection in the real axis. … Magnify 3D Help

… ►

…

32: 19.30 Lengths of Plane Curves

§19.30 Lengths of Plane Curves

►

§19.30(i) Ellipse

… ►

§19.30(ii) Hyperbola

… ►For other plane curves with arclength representable by an elliptic integral see Greenhill (1892, p. 190) and Bowman (1953, pp. 32–33).

33: 15.17 Mathematical Applications

… ►

§15.17(ii) Conformal Mappings

►The quotient of two solutions of (15.10.1) maps the closed upper half-plane

\Im z\geq 0

conformally onto a curvilinear triangle. …Hypergeometric functions, especially complete elliptic integrals, also play an important role in quasiconformal mapping. … ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL

(2,\mathbb{R})

, and spherical functions on certain nonsymmetric Gelfand pairs. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …

34: 19.39 Software

… ►Unless otherwise stated, the functions are

K\left(k\right)

and

E\left(k\right)

, with

0\leq k^{2}(=m)\leq 1

. … ►For other software, sometimes with

\Pi\left(\alpha^{2},k\right)

and complex variables, see the Software Index. … ►Unless otherwise stated, the variables are real, and the functions are

F\left(\phi,k\right)

and

E\left(\phi,k\right)

. ►For research software see Bulirsch (1965b, function

\operatorname{el2}

), Bulirsch (1969b, function

\operatorname{el3}

), Jefferson (1961), and Neuman (1969a, functions

E\left(\phi,k\right)

and

\Pi\left(\phi,k^{2},k\right)

). … ►

§19.39(iv) Symmetric Integrals

…

35: 19 Elliptic Integrals

Chapter 19 Elliptic Integrals

…

36: 35.2 Laplace Transform

… ►For any complex symmetric matrix

\mathbf{Z}

, … ►Suppose there exists a constant

\mathbf{X}_{0}\in{\boldsymbol{\Omega}}

such that

|f(\mathbf{X})|<\operatorname{etr}\left(-\mathbf{X}_{0}\mathbf{X}\right)

for all

\mathbf{X}\in{\boldsymbol{\Omega}}

. Then (35.2.1) converges absolutely on the region

\Re\left(\mathbf{Z}\right)>\mathbf{X}_{0}

, and

g(\mathbf{Z})

is a complex analytic function of all elements

z_{j,k}

\mathbf{Z}

. … ►Assume that

\int_{\boldsymbol{\mathcal{S}}}\left|g(\mathbf{U}+\mathrm{i}\mathbf{V})\right|% \,\mathrm{d}{\mathbf{V}}

converges, and also that its limit as

\mathbf{U}\to\infty

0

. …where the integral is taken over all

\mathbf{Z}=\mathbf{U}+\mathrm{i}\mathbf{V}

such that

\mathbf{U}>\mathbf{X}_{0}

and

\mathbf{V}

ranges over

\boldsymbol{\mathcal{S}}

. …

37: 22.13 Derivatives and Differential Equations

… ►

§22.13(i) Derivatives

… ►For alternative, and symmetric, formulations of these results see Carlson (2004, 2006a). ►

§22.13(ii) First-Order Differential Equations

… ►For alternative, and symmetric, formulations of these results see Carlson (2006a). … ►For alternative, and symmetric, formulations of these results see Carlson (2006a).

38: 19.29 Reduction of General Elliptic Integrals

§19.29 Reduction of General Elliptic Integrals

►

§19.29(i) Reduction Theorems

►These theorems reduce integrals over a real interval

(y,x)

of certain integrands containing the square root of a quartic or cubic polynomial to symmetric integrals over

(0,\infty)

containing the square root of a cubic polynomial (compare §19.16(i)). … ►which shows how to express the basic integral

I(-\mathbf{e}_{j})

in terms of symmetric integrals by using (19.29.4) and either (19.29.7) or (19.29.8). … ►It can be expressed in terms of symmetric integrals by setting

a_{5}=1

and

b_{5}=0

in (19.29.8). …

39: 19.26 Addition Theorems

§19.26 Addition Theorems

… ►where

\mu>0

and …where

\lambda>0

y>0

x\geq 0

, and … ►

§19.26(ii) Case $x=0$

… ►

§19.26(iii) Duplication Formulas

…

40: 19.23 Integral Representations

§19.23 Integral Representations

►In (19.23.1)–(19.23.3) we assume

\Re y>0

and

\Re z>0

. ►

19.23.1

R_{F}\left(0,y,z\right)=\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^{2}\theta)^% {-1/2}\,\mathrm{d}\theta,

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : sine function
Referenced by:: §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §19.23 and Ch.19