general elliptic integrals

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31: Software Index

	Open Source									With Book		Commercial
…
4.48(iii) General Precision	✓			✓	✓	✓		✓	✓				✓	✓	a		REDUCE
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19 Elliptic Integrals
19.39(ii) $K\left(k\right)$ , $E\left(k\right)$ , $0\leq k^{2}\leq 1$	✓	✓	✓	✓		✓	✓	✓	✓			✓	✓	✓	✓
19.39(iii) $F\left(\phi,k\right)$ , $E\left(\phi,k\right)$ , $\phi,k\in\mathbb{R}$	✓	✓	✓	✓		✓	✓	✓	✓		✓		✓	✓	a	✓
…
22 Jacobian Elliptic Functions
…

…

32: 19.31 Probability Distributions

§19.31 Probability Distributions

►

R_{G}\left(x,y,z\right)

and

R_{F}\left(x,y,z\right)

occur as the expectation values, relative to a normal probability distribution in

{\mathbb{R}}^{2}

{\mathbb{R}}^{3}

, of the square root or reciprocal square root of a quadratic form. More generally, let

\mathbf{A}

(

=[a_{r,s}]

) and

\mathbf{B}

(

=[b_{r,s}]

) be real positive-definite matrices with

n

rows and

n

columns, and let

\lambda_{1},\dots,\lambda_{n}

be the eigenvalues of

\mathbf{A}\mathbf{B}^{-1}

. …§19.16(iii) shows that for

n=3

the incomplete cases of

R_{F}

and

R_{G}

occur when

\mu=-1/2

and

\mu=1/2

, respectively, while their complete cases occur when

n=2

. ►For (19.31.2) and generalizations see Carlson (1972b).

33: 19.27 Asymptotic Approximations and Expansions

§19.27 Asymptotic Approximations and Expansions

… ►

§19.27(ii) $R_{F}\left(x,y,z\right)$

… ►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)). … ►A similar (but more general) situation prevails for

R_{-a}\left(\mathbf{b};\mathbf{z}\right)

when some of the variables

z_{1},\dots,z_{n}

are smaller in magnitude than the rest; see Carlson (1985, (4.16)–(4.19) and (2.26)–(2.29)).

34: 22.2 Definitions

§22.2 Definitions

… ►where

K\left(k\right)

{K^{\prime}}\left(k\right)

are defined in §19.2(ii). … ►As a function of

z

, with fixed

k

, each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … ►

\operatorname{ss}\left(z,k\right)=1

. …

35: 23.15 Definitions

… ►

§23.15(i) General Modular Functions

… ►

23.15.1

q=\exp\left(-\pi\frac{{K^{\prime}}\left(k\right)}{K\left(k\right)}\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, ${K^{\prime}}\left(\NVar{k}\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $\exp\NVar{z}$ : exponential function, $q$ : nome and $k$ : modulus
Permalink:: http://dlmf.nist.gov/23.15.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §23.15(i), §23.15 and Ch.23

… ►

Elliptic Modular Function

►

23.15.6

\lambda\left(\tau\right)=\frac{{\theta_{2}}^{4}\left(0,q\right)}{{\theta_{3}}^% {4}\left(0,q\right)};

ⓘ

Defines:: $\lambda\left(\NVar{\tau}\right)$ : elliptic modular function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome and $\tau$ : complex variable
Referenced by:: §20.9(ii), §23.17(iii)
Permalink:: http://dlmf.nist.gov/23.15.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §23.15(ii), §23.15(ii), §23.15 and Ch.23

…

36: 22.16 Related Functions

… ►

§22.16(i) Jacobi’s Amplitude ( $\operatorname{am}$ ) Function

… ►

Integral Representation

… ►

Relation to Elliptic Integrals

… ►See Figure 22.16.2. … ►

Relation to the Elliptic Integral $E\left(\phi,k\right)$

…

37: 36.5 Stokes Sets

… ►

§36.5(ii) Cuspoids

… ►They generate a pair of cusp-edged sheets connected to the cusped sheets of the swallowtail bifurcation set (§36.4). … ►The first sheet corresponds to

x<0

and is generated as a solution of Equations (36.5.6)–(36.5.9). … ►

§36.5(iii) Umbilics

►

Elliptic Umbilic Stokes Set (Codimension three)

…

38: 19.10 Relations to Other Functions

… ►

§19.10(i) Theta and Elliptic Functions

►For relations of Legendre’s integrals to theta functions, Jacobian functions, and Weierstrass functions, see §§20.9(i), 22.15(ii), and 23.6(iv), respectively. … ►

§19.10(ii) Elementary Functions

►If

y>0

is assumed (without loss of generality), then …In each case when

y=1

, the quantity multiplying

R_{C}

supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0. …

39: 22.9 Cyclic Identities

§22.9 Cyclic Identities

►

§22.9(i) Notation

… ► … ► … ►

40: 23.5 Special Lattices

… ►

§23.5(ii) Rectangular Lattice

… ►

§23.5(iii) Lemniscatic Lattice

… ►

§23.5(iv) Rhombic Lattice

… ►As a function of

\Im e_{3}

the root

e_{1}

is increasing. … ►

§23.5(v) Equianharmonic Lattice

…