DLMF: Untitled Document

… ► ►►►

$x, y$	real variables.
…
$k^{\prime}$	complementary modulus, $k^{2}+{k^{\prime}}^{2}=1$ . If $k\in[0,1]$ , then $k^{\prime}\in[0,1]$ .
…

… ►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)

. … ►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). …

… ►Also, let

\mathbf{Z}

be an arbitrary

g\times h

matrix. … ►

21.6.3

\prod_{j=1}^{h}\theta\left(\sum_{k=1}^{h}T_{jk}\mathbf{z}_{k}\middle|% \boldsymbol{{\Omega}}\right)=\frac{1}{\mathcal{D}^{g}}\sum_{\mathbf{A}\in% \mathcal{K}}\sum_{\mathbf{B}\in\mathcal{K}}e^{2\pi i\operatorname{tr}\left[% \frac{1}{2}\mathbf{A}^{\mathrm{T}}\boldsymbol{{\Omega}}\mathbf{A}+\mathbf{A}^{% \mathrm{T}}[\mathbf{Z}+\mathbf{B}]\right]}\*\prod_{j=1}^{h}\theta\left(\mathbf% {z}_{j}+\boldsymbol{{\Omega}}\mathbf{a}_{j}+\mathbf{b}_{j}\middle|\boldsymbol{% {\Omega}}\right),

►where

\mathbf{z}_{j}

,

\mathbf{a}_{j}

,

\mathbf{b}_{j}

denote respectively the

j

th columns of

\mathbf{Z}

,

\mathbf{A}

,

\mathbf{B}

. … ►

21.6.5

\mathbf{T}=\frac{1}{2}\begin{bmatrix}1&1&1&1\\ 1&1&-1&-1\\ 1&-1&1&-1\\ 1&-1&-1&1\end{bmatrix}.

ⓘ

Referenced by:: Erratum (V1.0.17) for Equation (21.6.5)
Permalink:: http://dlmf.nist.gov/21.6.E5
Encodings:: pMML, png, TeX
Errata (effective with 1.0.17):: Originally the prefactor $\frac{1}{2}$ on the right-hand side was missing.
Reported 2017-08-12 by Wolfgang Bauhardt
See also:: Annotations for §21.6(i), §21.6(i), §21.6 and Ch.21

… ►

21.6.7

\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}+% \mathbf{c}_{3}+\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}+% \mathbf{d}_{3}+\mathbf{d}_{4}]}\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{u}+% \mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{% }{\tfrac{1}{2}[\mathbf{c}_{1}+\mathbf{c}_{2}-\mathbf{c}_{3}-\mathbf{c}_{4}]}{% \tfrac{1}{2}[\mathbf{d}_{1}+\mathbf{d}_{2}-\mathbf{d}_{3}-\mathbf{d}_{4}]}% \left(\frac{\mathbf{x}+\mathbf{y}-\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol% {{\Omega}}\right)\*\theta\genfrac{[}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-% \mathbf{c}_{2}+\mathbf{c}_{3}-\mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-% \mathbf{d}_{2}+\mathbf{d}_{3}-\mathbf{d}_{4}]}\left(\frac{\mathbf{x}-\mathbf{y% }+\mathbf{u}-\mathbf{v}}{2}\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{% [}{]}{0.0pt}{}{\tfrac{1}{2}[\mathbf{c}_{1}-\mathbf{c}_{2}-\mathbf{c}_{3}+% \mathbf{c}_{4}]}{\tfrac{1}{2}[\mathbf{d}_{1}-\mathbf{d}_{2}-\mathbf{d}_{3}+% \mathbf{d}_{4}]}\left(\frac{\mathbf{x}-\mathbf{y}-\mathbf{u}+\mathbf{v}}{2}% \middle|\boldsymbol{{\Omega}}\right)\\ =\frac{1}{2^{g}}\sum_{\boldsymbol{{\alpha}}\in\frac{1}{2}{\mathbb{Z}}^{g}/{% \mathbb{Z}}^{g}}\,\sum_{\boldsymbol{{\beta}}\in\frac{1}{2}{\mathbb{Z}}^{g}/{% \mathbb{Z}}^{g}}e^{-2\pi i\boldsymbol{{\beta}}\cdot[\mathbf{c}_{1}+\mathbf{c}_% {2}+\mathbf{c}_{3}+\mathbf{c}_{4}]}\*\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {1}+\boldsymbol{{\alpha}}}{\mathbf{d}_{1}+\boldsymbol{{\beta}}}\left(\mathbf{x% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {2}+\boldsymbol{{\alpha}}}{\mathbf{d}_{2}+\boldsymbol{{\beta}}}\left(\mathbf{y% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {3}+\boldsymbol{{\alpha}}}{\mathbf{d}_{3}+\boldsymbol{{\beta}}}\left(\mathbf{u% }\middle|\boldsymbol{{\Omega}}\right)\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{c}_% {4}+\boldsymbol{{\alpha}}}{\mathbf{d}_{4}+\boldsymbol{{\beta}}}\left(\mathbf{v% }\middle|\boldsymbol{{\Omega}}\right).

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.6.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §21.6(i), §21.6(i), §21.6 and Ch.21

…

… ►

Table 18.8.1: Classical OP’s: differential equations

A(x)f^{\prime\prime}(x)+B(x)f^{\prime}(x)+C(x)f(x)+\lambda_{n}f(x)=0

.

► ►►►►►►

#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	$\lambda_{n}$
…
2	$\begin{gathered}\textstyle\left(\sin\frac{1}{2}x\right)^{\alpha+\frac{1}{2}}% \left(\cos\frac{1}{2}x\right)^{\beta+\frac{1}{2}}\\ \times P^{(\alpha,\beta)}_{n}\left(\cos x\right)\end{gathered}$	$1$	$0$	$\dfrac{\tfrac{1}{4}-\alpha^{2}}{4{\sin}^{2}\frac{1}{2}x}+\dfrac{\frac{1}{4}-% \beta^{2}}{4{\cos}^{2}\frac{1}{2}x}$	$\left(n+\frac{1}{2}(\alpha+\beta+1)\right)^{2}$
…
9	${\mathrm{e}}^{-\frac{1}{2}x^{2}}x^{\alpha+\frac{1}{2}}L^{(\alpha)}_{n}\left(x^% {2}\right)$	$1$	$0$	$-x^{2}+(\frac{1}{4}-\alpha^{2})x^{-2}$	$4n+2\alpha+2$
…
12	$H_{n}\left(x\right)$	$1$	$-2x$	$0$	$2n$
13	${\mathrm{e}}^{-\frac{1}{2}x^{2}}H_{n}\left(x\right)$	$1$	$0$	$-x^{2}$	$2n+1$
…

► ►Item 11 of Table 18.8.1 yields (18.39.36) for

Z=1

.

… ►Results of low (

~{}2

to

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

to

20

. … ►Convergence is

\sim O\left(N^{-2}\right)

. … ►Here

x(t,N)

is an interpolation of the abscissas

x_{i,N},i=1,2,\dots,N

, that is,

x(i,N)=x_{i,N}

, allowing differentiation by

i

. …where the coefficients are defined recursively via

a_{1}=\frac{x_{1,N}}{x_{2,N}}-1

, and …The PWCF

x(t,N)

is a minimally oscillatory algebraic interpolation of the abscissas

x_{i,N},i=1,2,\dots,N

. …

… ►

35.5.1

A_{\nu}\left(\boldsymbol{{0}}\right)=\frac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(% m+1)\right)},

\nu\in\mathbb{C}

.

ⓘ

Symbols:: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), $\mathbb{C}$ : complex plane, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.5.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.5(i), §35.5 and Ch.35

►

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

… ►

35.5.5

\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

…

… ►

Á. Elbert and A. Laforgia (1994) Interlacing properties of the zeros of Bessel functions. Atti Sem. Mat. Fis. Univ. Modena XLII (2), pp. 525–529.

ⓘ

External Links:: ISSN 0041-8986, MathReview (Evangelos Ifantis), ZentralBlatt
Referenced by:: §10.21(i).
Encodings:: BibTeX

►

Á. Elbert and A. Laforgia (1997) An upper bound for the zeros of the derivative of Bessel functions. Rend. Circ. Mat. Palermo (2) 46 (1), pp. 123–130.

ⓘ

External Links:: ISSN 0009-725X, MathReview (Boro Döring), ZentralBlatt
Referenced by:: §10.21(vii).
Encodings:: BibTeX

… ►

E. B. Elliott (1903) A formula including Legendre’s

EK^{\prime}+KE^{\prime}-KK^{\prime}=\frac{1}{2}\pi

. Messenger of Math. 33, pp. 31–32.

ⓘ

Notes:: Errata on p. 45 of the same volume.
External Links:: ZentralBlatt
Referenced by:: §15.16.
Encodings:: BibTeX

… ►

F. H. L. Essler, H. Frahm, A. R. Its, and V. E. Korepin (1996) Painlevé transcendent describes quantum correlation function of the

X X Z

antiferromagnet away from the free-fermion point. J. Phys. A 29 (17), pp. 5619–5626.

ⓘ

External Links:: ISSN 0305-4470, Document, MathReview (Estelle L. Basor), ZentralBlatt
Referenced by:: §32.16.
Encodings:: BibTeX

… ►

W. N. Everitt (1982) On the transformation theory of ordinary second-order linear symmetric differential expressions. Czechoslovak Math. J. 32(107) (2), pp. 275–306.

ⓘ

Notes:: With a loose Russian summary
External Links:: ISSN 0011-4642, MathReview (D. Hinton)
Referenced by:: §1.13(viii).
Encodings:: BibTeX

…

… ►

35.6.1

{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}{k!% }\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}}{{\left[b\right]_{\kappa}}}Z% _{\kappa}\left(\mathbf{T}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $!$ : factorial (as in $n!$ ), ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $k$ : nonnegative integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(i), §35.6 and Ch.35

►

35.6.2

\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}+\mathbf{X}\right|}^{% b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},

\Re\left(a\right)>\frac{1}{2}(m-1)

,

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

.

ⓘ

Defines:: $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind)
Symbols:: $\in$ : element of, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(i), §35.6 and Ch.35

… ►

35.6.3

L^{(\gamma)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+\nu+% \frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*{{}_% {1}F_{1}}\left({-\nu\atop\gamma+\frac{1}{2}(m+1)};\mathbf{T}\right),

\Re\left(\gamma\right),\Re\left(\gamma+\nu\right)>-1

.

ⓘ

Defines:: $L^{(\NVar{\gamma})}_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Laguerre function of matrix argument
Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $\mathbf{T}$ : real symmetric $m\times m$ matrix and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(i), §35.6(i), §35.6 and Ch.35

… ►

35.6.6

\mathrm{B}_{m}\left(b_{1},b_{2}\right)\left|\mathbf{T}\right|^{b_{1}+b_{2}-% \frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}+a_{2}\atop b_{1}+b_{2}};\mathbf{T}% \right)=\int_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{T}}\left|\mathbf{X}\right|^{% b_{1}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}\left({a_{1}\atop b_{1}};\mathbf{X}\right)% {\left|\mathbf{T}-\mathbf{X}\right|}^{b_{2}-\frac{1}{2}(m+1)}{{}_{1}F_{1}}% \left({a_{2}\atop b_{2}};\mathbf{T}-\mathbf{X}\right)\,\mathrm{d}{\mathbf{X}},

\Re\left(b_{1}\right),\Re\left(b_{2}\right)>\frac{1}{2}(m-1)

.

… ►

35.6.8

\int_{\boldsymbol{\Omega}}\left|\mathbf{T}\right|^{c-\frac{1}{2}(m+1)}\Psi% \left(a;b;\mathbf{T}\right)\,\mathrm{d}{\mathbf{T}}=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(a-c\right)\Gamma_{m}\left(c-b+\frac{1}{2}(m+1)\right)}{% \Gamma_{m}\left(a\right)\Gamma_{m}\left(a-b+\frac{1}{2}(m+1)\right)},

\Re\left(a\right)>\Re\left(c\right)+\frac{1}{2}(m-1)>m-1

,

\Re\left(c-b\right)>-1

.

ⓘ

Symbols:: $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind), $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left|\NVar{\mathbf{X}}\right|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable, $c$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.6.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(ii), §35.6 and Ch.35

…

… ►

R. B. Paris (2005a) A Kummer-type transformation for a

{}_{2}F_{2}

hypergeometric function. J. Comput. Appl. Math. 173 (2), pp. 379–382.

ⓘ

External Links:: ISSN 0377-0427, Document, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §16.6.
Encodings:: BibTeX

… ►

J. B. Parkinson (1969) Optical properties of layer antiferromagnets with

\mathrm{K}_{2}\mathrm{NiF}_{4}

structure. J. Phys. C: Solid State Physics 2 (11), pp. 2012–2021.

ⓘ

External Links:: Document
Referenced by:: §19.35(ii).
Encodings:: BibTeX

►

R. Parnes (1972) Complex zeros of the modified Bessel function

K_{n}(Z)

. Math. Comp. 26 (120), pp. 949–953.

ⓘ

External Links:: ISSN 0025-5718, FullText, MathReview, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

… ►

B. Pichon (1989) Numerical calculation of the generalized Fermi-Dirac integrals. Comput. Phys. Comm. 55 (2), pp. 127–136.

ⓘ

External Links:: Document
Referenced by:: §25.18(i).
Encodings:: BibTeX

… ►

J. L. Powell (1947) Recurrence formulas for Coulomb wave functions. Physical Rev. (2) 72 (7), pp. 626–627.

ⓘ

External Links:: Document, MathReview (S. C. van Veen), ZentralBlatt
Referenced by:: §33.4.
Encodings:: BibTeX

…

… ►and

\mathscr{Z}_{p}

is any linear combination of the modified Bessel functions

I_{p}

and

e^{p\pi\mathrm{i}}K_{p}

(§10.25(ii)). … ►The function on the right-hand side is recessive in the sector

-(2j-1)\pi/m\leq\operatorname{ph}z\leq(2j+1)\pi/m

, and is therefore an essential member of any numerically satisfactory pair of solutions in this region. … ►where

\alpha>-2

and

x>0

. … ►The integration paths

\mathscr{L}_{0}

,

\mathscr{L}_{1}

,

\mathscr{L}_{2}

,

\mathscr{L}_{3}

are depicted in Figure 9.13.1.

\mathscr{I}_{1}

,

\mathscr{I}_{2}

,

\mathscr{I}_{3}

are depicted in Figure 9.13.2. …

… ►

§35.8(iii) ${{}_{3}F_{2}}$ Case

►

Kummer Transformation

►Let

c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}

. … ►Let

a_{1}+a_{2}+a_{3}+\frac{1}{2}(m+1)=b_{1}+b_{2}

; one of the

a_{j}

be a negative integer;

\Re\left(b_{1}-a_{1}\right)

,

\Re\left(b_{1}-a_{2}\right)

,

\Re\left(b_{1}-a_{3}\right)

,

\Re\left(b_{1}-a_{1}-a_{2}-a_{3}\right)>\frac{1}{2}(m-1)

. … ►Again, let

c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}

. …

SL(2,Z)

21—30 of 803 matching pages

21: 22.1 Special Notation

22: 21.6 Products

23: 18.8 Differential Equations

24: 18.40 Methods of Computation

25: 35.5 Bessel Functions of Matrix Argument

26: Bibliography E

27: 35.6 Confluent Hypergeometric Functions of Matrix Argument

28: Bibliography P

29: 9.13 Generalized Airy Functions

30: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8(iii) ${{}_{3}F_{2}}$ Case

Kummer Transformation

SL(2,Z)

21—30 of 803 matching pages

21: 22.1 Special Notation

22: 21.6 Products

23: 18.8 Differential Equations

24: 18.40 Methods of Computation

25: 35.5 Bessel Functions of Matrix Argument

26: Bibliography E

27: 35.6 Confluent Hypergeometric Functions of Matrix Argument

28: Bibliography P

29: 9.13 Generalized Airy Functions

30: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8(iii) F 2 3 Case

Kummer Transformation

§35.8(iii) ${{}_{3}F_{2}}$ Case