DLMF: Untitled Document

… ► ►►►►►►

$a, b$	complex variables.
…
$\boldsymbol{\mathcal{S}}$	space of all real symmetric matrices.
$\mathbf{S},\mathbf{T},\mathbf{X}$	real symmetric matrices.
…
${\boldsymbol{\Omega}}$	space of positive-definite real symmetric matrices.
…
$\mathbf{Z}$	complex symmetric matrix.
…

… ►Related notations for the Bessel functions are

\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)

(Faraut and Korányi (1994, pp. 320–329)),

K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)

(Terras (1988, pp. 49–64)), and

\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)

(Faraut and Korányi (1994, pp. 357–358)).

§19.31 Probability Distributions

…

§22.8 Addition Theorems

… ►

§22.8(iii) Special Relations Between Arguments

… ►

22.8.23

\begin{vmatrix}\operatorname{sn}z_{1}\operatorname{cn}z_{1}&\operatorname{cn}z% _{1}\operatorname{dn}z_{1}&\operatorname{cn}z_{1}&\operatorname{dn}z_{1}\\ \operatorname{sn}z_{2}\operatorname{cn}z_{2}&\operatorname{cn}z_{2}% \operatorname{dn}z_{2}&\operatorname{cn}z_{2}&\operatorname{dn}z_{2}\\ \operatorname{sn}z_{3}\operatorname{cn}z_{3}&\operatorname{cn}z_{3}% \operatorname{dn}z_{3}&\operatorname{cn}z_{3}&\operatorname{dn}z_{3}\\ \operatorname{sn}z_{4}\operatorname{cn}z_{4}&\operatorname{cn}z_{4}% \operatorname{dn}z_{4}&\operatorname{cn}z_{4}&\operatorname{dn}z_{4}\end{% vmatrix}=0.

ⓘ

Symbols:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $\det$ : determinant, $z$ : complex and $k$ : modulus
Permalink:: http://dlmf.nist.gov/22.8.E23
Encodings:: pMML, png, TeX
See also:: Annotations for §22.8(iii), §22.8 and Ch.22

… ►If sums/differences of the

z_{j}

’s are rational multiples of

K\left(k\right)

, then further relations follow. …

… ►

35.5.2

A_{\nu}\left(\mathbf{T}\right)=A_{\nu}\left(\boldsymbol{{0}}\right)\sum_{k=0}^% {\infty}\frac{(-1)^{k}}{k!}\sum_{|\kappa|=k}\frac{1}{{\left[\nu+\frac{1}{2}(m+% 1)\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right),

\nu\in\mathbb{C}

,

\mathbf{T}\in\boldsymbol{\mathcal{S}}

.

ⓘ

Symbols:: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), $\mathbb{C}$ : complex plane, $\in$ : element of, $!$ : factorial (as in $n!$ ), ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\boldsymbol{\mathcal{S}}$ : space of real symmetric $m\times m$ matrices, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $k$ : nonnegative integer, $m$ : positive integer, $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.5.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §35.5(i), §35.5 and Ch.35

… ►

35.5.4

\int_{\boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)% \left|\mathbf{X}\right|^{\nu}A_{\nu}\left(\mathbf{S}\mathbf{X}\right)\,\mathrm% {d}{\mathbf{X}}=\operatorname{etr}\left(-\mathbf{S}\mathbf{T}^{-1}\right)\left% |\mathbf{T}\right|^{-\nu-\frac{1}{2}(m+1)},

\mathbf{S}\in\boldsymbol{\mathcal{S}}

,

\mathbf{T}\in{\boldsymbol{\Omega}}

;

\Re\left(\nu\right)>-1

.

►

35.5.5

\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}A_{\nu_{1}}\left(\mathbf{S% }_{1}\mathbf{X}\right)\left|\mathbf{X}\right|^{\nu_{1}}\*A_{\nu_{2}}\left(% \mathbf{S}_{2}(\mathbf{T}-\mathbf{X})\right)\left|\mathbf{T}-\mathbf{X}\right|% ^{\nu_{2}}\,\mathrm{d}{\mathbf{X}}=\left|\mathbf{T}\right|^{\nu_{1}+\nu_{2}+% \frac{1}{2}(m+1)}A_{\nu_{1}+\nu_{2}+\frac{1}{2}(m+1)}\left((\mathbf{S}_{1}+% \mathbf{S}_{2})\mathbf{T}\right),

\nu_{j}\in\mathbb{C}

,

\Re\left(\nu_{j}\right)>-1

,

j=1,2

;

\mathbf{S}_{1},\mathbf{S}_{2}\in\boldsymbol{\mathcal{S}}

;

\mathbf{T}\in{\boldsymbol{\Omega}}

.

… ►

35.5.7

\int_{\boldsymbol{\Omega}}A_{\nu_{1}}\left(\mathbf{T}\mathbf{X}\right)B_{-\nu_% {2}}\left(\mathbf{S}\mathbf{X}\right)\left|\mathbf{X}\right|^{\nu_{1}}\,% \mathrm{d}{\mathbf{X}}=\frac{1}{A_{\nu_{1}+\nu_{2}}\left(\boldsymbol{{0}}% \right)}\left|\mathbf{S}\right|^{\nu_{2}}\left|\mathbf{T}+\mathbf{S}\right|^{-% (\nu_{1}+\nu_{2}+\frac{1}{2}(m+1))},

\Re\left(\nu_{1}+\nu_{2}\right)>-1

;

\mathbf{S},\mathbf{T}\in{\boldsymbol{\Omega}}

.

►

35.5.8

\int_{\mathbf{O}(m)}\operatorname{etr}\left(\mathbf{S}\mathbf{H}\right)\mathrm% {d}{\mathbf{H}}=\frac{A_{-1/2}\left(-\frac{1}{4}\mathbf{S}\mathbf{S}^{\mathrm{% T}}\right)}{A_{-1/2}\left(\boldsymbol{{0}}\right)},

\mathbf{S}

arbitrary.

ⓘ

Symbols:: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\mathbf{S}$ : real symmetric $m\times m$ matrix, $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $\mathrm{d}{\mathbf{H}}$ : normalized Haar measure on $\mathbf{O}(m)$ , $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Referenced by:: §35.10, §35.9
Permalink:: http://dlmf.nist.gov/35.5.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §35.5(ii), §35.5 and Ch.35

…

… ►

§22.4(i) Distribution

… ► … ►Figure 22.4.1 illustrates the locations in the

z

-plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices

0

,

2K

,

2K+2iK^{\prime}

,

2iK^{\prime}

. … ► … ►Figure 22.4.2 depicts the fundamental unit cell in the

z

-plane, with vertices

\mbox{s}=0

,

\mbox{c}=K

,

\mbox{d}=K+iK^{\prime}

,

\mbox{n}=iK^{\prime}

. …

§22.21 Tables

►Spenceley and Spenceley (1947) tabulates

\operatorname{sn}\left(Kx,k\right)

,

\operatorname{cn}\left(Kx,k\right)

,

\operatorname{dn}\left(Kx,k\right)

,

\operatorname{am}\left(Kx,k\right)

,

\mathcal{E}\left(Kx,k\right)

for

\operatorname{arcsin}k=1^{\circ}(1^{\circ})89^{\circ}

and

x=0\left(\tfrac{1}{90}\right)1

to 12D, or 12 decimals of a radian in the case of

\operatorname{am}\left(Kx,k\right)

. ►Curtis (1964b) tabulates

\operatorname{sn}\left(mK/n,k\right)

,

\operatorname{cn}\left(mK/n,k\right)

,

\operatorname{dn}\left(mK/n,k\right)

for

n=2(1)15

,

m=1(1)n-1

, and

q

(not

k

)

=0(.005)0.35

to 20D. ►Lawden (1989, pp. 280–284 and 293–297) tabulates

\operatorname{sn}\left(x,k\right)

,

\operatorname{cn}\left(x,k\right)

,

\operatorname{dn}\left(x,k\right)

,

\mathcal{E}\left(x,k\right)

,

\mathrm{Z}\left(x|k\right)

to 5D for

k=0.1(.1)0.9

,

x=0(.1)X

, where

X

ranges from 1. … ►Zhang and Jin (1996, p. 678) tabulates

\operatorname{sn}\left(Kx,k\right)

,

\operatorname{cn}\left(Kx,k\right)

,

\operatorname{dn}\left(Kx,k\right)

for

k=\frac{1}{4},\frac{1}{2}

and

x=0(.1)4

to 7D. …

§22.17 Moduli Outside the Interval [0,1]

►

§22.17(i) Real or Purely Imaginary Moduli

►Jacobian elliptic functions with real moduli in the intervals

(-\infty,0)

and

(1,\infty)

, or with purely imaginary moduli are related to functions with moduli in the interval

[0,1]

by the following formulas. … ►

§22.17(ii) Complex Moduli

… ►For proofs of these results and further information see Walker (2003).

… ►

§23.6(ii) Jacobian Elliptic Functions

… ►

§23.6(iii) General Elliptic Functions

… ►

§23.6(iv) Elliptic Integrals

… ►For relations to symmetric elliptic integrals see §19.25(vi). … ►

… ► ►►►

$x, y$	real variables.
…
$K$ , ${K^{\prime}}$	$K\left(k\right)$ , ${K^{\prime}}\left(k\right)=K\left(k^{\prime}\right)$ (complete elliptic integrals of the first kind (§19.2(ii))).
…

… ►The functions treated in this chapter are the three principal Jacobian elliptic functions

\operatorname{sn}\left(z,k\right)

,

\operatorname{cn}\left(z,k\right)

,

\operatorname{dn}\left(z,k\right)

; the nine subsidiary Jacobian elliptic functions

\operatorname{cd}\left(z,k\right)

,

\operatorname{sd}\left(z,k\right)

,

\operatorname{nd}\left(z,k\right)

,

\operatorname{dc}\left(z,k\right)

,

\operatorname{nc}\left(z,k\right)

,

\operatorname{sc}\left(z,k\right)

,

\operatorname{ns}\left(z,k\right)

,

\operatorname{ds}\left(z,k\right)

,

\operatorname{cs}\left(z,k\right)

; the amplitude function

\operatorname{am}\left(x,k\right)

; Jacobi’s epsilon and zeta functions

\mathcal{E}\left(x,k\right)

and

\mathrm{Z}\left(x|k\right)

. ►The notation

\operatorname{sn}\left(z,k\right)

,

\operatorname{cn}\left(z,k\right)

,

\operatorname{dn}\left(z,k\right)

is due to Gudermann (1838), following Jacobi (1827); that for the subsidiary functions is due to Glaisher (1882). Other notations for

\operatorname{sn}\left(z,k\right)

are

\mathrm{sn}(z\mathpunct{|}m)

and

\mathrm{sn}(z,m)

with

m=k^{2}

; see Abramowitz and Stegun (1964) and Walker (1996). …

… ►For example, at

z=K+iK^{\prime}

,

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

,

\ifrac{\mathrm{d}\operatorname{sn}\left(z,k\right)}{\mathrm{d}z}=0

. … ►

§22.5(ii) Limiting Values of $k$

►If

k\to 0+

, then

K\to\pi/2

and

K^{\prime}\to\infty

; if

k\to 1-

, then

K\to\infty

and

K^{\prime}\to\pi/2

. … ►Expansions for

K,K^{\prime}

as

k\to 0

or

1

are given in §§19.5, 19.12. … ►

symmetric elliptic%0Aintegrals

41—50 of 738 matching pages

41: 35.1 Special Notation

42: 19.31 Probability Distributions

§19.31 Probability Distributions

43: 22.8 Addition Theorems

§22.8 Addition Theorems

§22.8(iii) Special Relations Between Arguments

44: 35.5 Bessel Functions of Matrix Argument

45: 22.4 Periods, Poles, and Zeros

§22.4(i) Distribution

46: 22.21 Tables

§22.21 Tables

47: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

§22.17(i) Real or Purely Imaginary Moduli

§22.17(ii) Complex Moduli

48: 23.6 Relations to Other Functions

§23.6(ii) Jacobian Elliptic Functions

§23.6(iii) General Elliptic Functions

§23.6(iv) Elliptic Integrals

49: 22.1 Special Notation

50: 22.5 Special Values

§22.5(ii) Limiting Values of $k$

symmetric elliptic%0Aintegrals

41—50 of 738 matching pages

41: 35.1 Special Notation

42: 19.31 Probability Distributions

§19.31 Probability Distributions

43: 22.8 Addition Theorems

§22.8 Addition Theorems

§22.8(iii) Special Relations Between Arguments

44: 35.5 Bessel Functions of Matrix Argument

45: 22.4 Periods, Poles, and Zeros

§22.4(i) Distribution

46: 22.21 Tables

§22.21 Tables

47: 22.17 Moduli Outside the Interval [0,1]

§22.17 Moduli Outside the Interval [0,1]

§22.17(i) Real or Purely Imaginary Moduli

§22.17(ii) Complex Moduli

48: 23.6 Relations to Other Functions

§23.6(ii) Jacobian Elliptic Functions

§23.6(iii) General Elliptic Functions

§23.6(iv) Elliptic Integrals

49: 22.1 Special Notation

50: 22.5 Special Values

§22.5(ii) Limiting Values of k

§22.5(ii) Limiting Values of $k$