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1: DLMF Project News

error generating summary

2: 15.20 Software

… ►

§15.20(ii) Real Parameters and Argument

… ►

§15.20(iii) Complex Parameters and Argument

…

3: 35.10 Methods of Computation

… ►See Yan (1992) for the

{{}_{1}F_{1}}

and

{{}_{2}F_{1}}

functions of matrix argument in the case

m=2

, and Bingham et al. (1992) for Monte Carlo simulation on

\mathbf{O}(m)

applied to a generalization of the integral (35.5.8). …

4: 15.4 Special Cases

… ►

§15.4(ii) Argument Unity

… ►

§15.4(iii) Other Arguments

…

5: 4.29 Graphics

… ►

§4.29(i) Real Arguments

… ►

See accompanying text — Figure 4.29.4: Principal values of $\operatorname{arctanh}x$ and $\operatorname{arccoth}x$ . ( $\operatorname{arctanh}x$ is complex when $x<-1$ or $x>1$ , and $\operatorname{arccoth}x$ is complex when $-1<x<1$ .) Magnify

… ►

►

§4.29(ii) Complex Arguments

►The conformal mapping

w=\sinh z

is obtainable from Figure 4.15.7 by rotating both the

w

-plane and the

z

-plane through an angle

\frac{1}{2}\pi

, compare (4.28.8). …

6: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►

35.7.1

{{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{|\kappa|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},

-c+\frac{1}{2}(j+1)\notin\mathbb{N}

1\leq j\leq m

;

\|\mathbf{T}\|<1

… ►

35.7.2

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

\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}

;

\gamma,\delta,\nu\in\mathbb{C}

;

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

ⓘ

Defines:: $P^{(\NVar{\gamma},\NVar{\delta})}_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Jacobi 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, $\mathbb{C}$ : complex plane, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.7.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §35.7(i), §35.7(i), §35.7 and Ch.35

… ►

35.7.3

{{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\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, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable and $k$ : nonnegative integer
Referenced by:: Erratum (V1.0.25) for Equation (35.7.3)
Permalink:: http://dlmf.nist.gov/35.7.E3
Encodings:: pMML, png, TeX
Notational change (effective with 1.0.25):: Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.
See also:: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

… ►Let

f:{\boldsymbol{\Omega}}\to\mathbb{C}

(a) be orthogonally invariant, so that

f(\mathbf{T})

is a symmetric function of

t_{1},\dots,t_{m}

, the eigenvalues of the matrix argument

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

; (b) be analytic in

t_{1},\dots,t_{m}

in a neighborhood of

\mathbf{T}=\boldsymbol{{0}}

; (c) satisfy

f(\boldsymbol{{0}})=1

. … ►Systems of partial differential equations for the

{{}_{0}F_{1}}

(defined in §35.8) and

{{}_{1}F_{1}}

functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …

7: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►The generalized hypergeometric function

{{}_{p}F_{q}}

with matrix argument

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

, numerator parameters

a_{1},\dots,a_{p}

, and denominator parameters

b_{1},\dots,b_{q}

is … ►

35.8.4

A_{\nu}\left(\mathbf{T}\right)=\dfrac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(m+1)% \right)}{{}_{0}F_{1}}\left({-\atop\nu+\frac{1}{2}(m+1)};-\mathbf{T}\right),

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

ⓘ

Symbols:: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ 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, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\boldsymbol{\mathcal{S}}$ : space of real symmetric $m\times m$ matrices, $\mathbf{T}$ : real symmetric $m\times m$ matrix and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(ii), §35.8 and Ch.35

… ►

35.8.8

{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\boldsymbol{{0}}% \right)=1.

ⓘ

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, $a$ : complex variable, $b$ : complex variable, $p$ : nonnegative integer, $q$ : nonnegative integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.8.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

… ►Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions

{{}_{p}F_{q}}

and

{{}_{p+1}F_{p}}

of matrix argument. A similar result for the

{{}_{0}F_{1}}

function of matrix argument is given in Faraut and Korányi (1994, p. 346). …

8: 35.6 Confluent Hypergeometric Functions of Matrix Argument

… ►

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}}\mathrm{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, $\mathrm{etr}\left(\NVar{\mathbf{X}}\right)$ : 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

… ►

35.6.7

{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\mathrm{etr}\left(\mathbf{T}% \right){{}_{1}F_{1}}\left({b-a\atop b};-\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, $\mathrm{etr}\left(\NVar{\mathbf{X}}\right)$ : exponential of trace, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix and $b$ : complex variable
Permalink:: http://dlmf.nist.gov/35.6.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(ii), §35.6 and Ch.35

… ►

35.6.9

\lim_{a\to\infty}{{}_{1}F_{1}}\left({a\atop\nu+\frac{1}{2}(m+1)};-a^{-1}% \mathbf{T}\right)=\frac{A_{\nu}\left(\mathbf{T}\right)}{A_{\nu}\left(% \boldsymbol{{0}}\right)}.

ⓘ

Symbols:: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ 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, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.6.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(iii), §35.6 and Ch.35

…

9: 14.2 Differential Equations

… ►Unless stated otherwise in §§14.2–14.20 it is assumed that the arguments of the functions

\mathsf{P}^{\mu}_{\nu}\left(x\right)

and

\mathsf{Q}^{\mu}_{\nu}\left(x\right)

lie in the interval

(-1,1)

, and the arguments of the functions

P^{\mu}_{\nu}\left(x\right)

Q^{\mu}_{\nu}\left(x\right)

, and

\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)

lie in the interval

(1,\infty)

. …

10: 10.17 Asymptotic Expansions for Large Argument

… ►

10.17.4

Y_{\nu}\left(z\right)\sim\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}\*\left(% \sin\omega\sum_{k=0}^{\infty}(-1)^{k}\frac{a_{2k}(\nu)}{z^{2k}}+\cos\omega\sum% _{k=0}^{\infty}(-1)^{k}\frac{a_{2k+1}(\nu)}{z^{2k+1}}\right),

|\operatorname{ph}z|\leq\pi-\delta

ⓘ

Symbols:: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant, $a_{k}(\nu)$ : polynomial coefficient and $\omega$
A&S Ref:: 9.2.6
Referenced by:: §10.17(iii), §10.17(iv), §10.18(iii), §10.24, §10.7(ii), §11.6(i)
Permalink:: http://dlmf.nist.gov/10.17.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §10.17(i), §10.17 and Ch.10

… ►

10.17.12

{H^{(2)}_{\nu}}'\left(z\right)\sim-i\left(\frac{2}{\pi z}\right)^{\frac{1}{2}}% e^{-i\omega}\sum_{k=0}^{\infty}(-i)^{k}\frac{b_{k}(\nu)}{z^{k}},

-2\pi+\delta\leq\operatorname{ph}z\leq\pi-\delta

ⓘ

Symbols:: ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $z$ : complex variable, $\nu$ : complex parameter, $\delta$ : small positive constant, $\omega$ and $b_{k}(\nu)$ : polynomial coefficient
A&S Ref:: 9.2.14
Referenced by:: §10.18(iii)
Permalink:: http://dlmf.nist.gov/10.17.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §10.17(ii), §10.17 and Ch.10

… ►

10.17.14

\left|R_{\ell}^{\pm}(\nu,z)\right|\leq 2|a_{\ell}(\nu)|\mathcal{V}_{z,\pm i% \infty}\left(t^{-\ell}\right)\*\exp\left(|\nu^{2}-\tfrac{1}{4}|\mathcal{V}_{z,% \pm i\infty}\left(t^{-1}\right)\right),

ⓘ

Defines:: $R_{\ell}^{\pm}(\nu,z)$ : remainder (locally)
Symbols:: $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\mathcal{V}_{\NVar{a,b}}\left(\NVar{f}\right)$ : total variation, $z$ : complex variable, $\nu$ : complex parameter and $a_{k}(\nu)$ : polynomial coefficient
Referenced by:: §10.17(iv), Erratum (V1.0.10) for Equation (10.17.14)
Permalink:: http://dlmf.nist.gov/10.17.E14
Encodings:: pMML, png, TeX
Errata (effective with 1.0.10):: Originally the factor $\mathcal{V}_{z,\pm i\infty}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as $\mathcal{V}_{z,\pm i\infty}\left(t^{-\ell}\right)$ .
Reported 2014-09-27 by Gergő Nemes
See also:: Annotations for §10.17(iv), §10.17 and Ch.10

…