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1: 35.1 Special Notation

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$a,b$	complex variables.
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$f(𝐗)$	complex-valued function with $𝐗\in 𝛀$.
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively ${\mathrm{\Gamma }}_m\left(a\right)$ and ${\mathrm{B}}_m(a,b)$, and the special functions of matrix argument: Bessel (of the first kind) $A_{\nu }\left(𝐓\right)$ and (of the second kind) $B_{\nu }\left(𝐓\right)$; confluent hypergeometric (of the first kind) ${}_1{}^{}F_1^{}(a;b;𝐓)$ or ${}_1{}^{}F_1^{}({\displaystyle \genfrac{}{}{0pt}{}{a}{b}};𝐓)$ and (of the second kind) $\mathrm{\Psi }(a;b;𝐓)$; Gaussian hypergeometric ${}_2{}^{}F_1^{}(a_1,a_2;b;𝐓)$ or ${}_2{}^{}F_1^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,a_2}{b}};𝐓)$; generalized hypergeometric ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}};𝐓)$. … ►Related notations for the Bessel functions are ${𝒥}_{\nu +\frac{1}{2}(m+1)}(𝐓)=A_{\nu }\left(𝐓\right)/A_{\nu }\left(𝟎\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_m(0,\mathrm{\dots },0,\nu |𝐒,𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Terras (1988, pp. 49–64)), and ${𝒦}_{\nu }(𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Faraut and Korányi (1994, pp. 357–358)).

2: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

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§35.5(i) Definitions

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§35.5(ii) Properties

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§35.5(iii) Asymptotic Approximations

►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).

3: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

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§35.8(i) Definition

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Convergence Properties

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Confluence

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Invariance

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4: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

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§35.6(i) Definitions

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Laguerre Form

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§35.6(ii) Properties

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§35.6(iii) Relations to Bessel Functions of Matrix Argument

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5: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

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§35.7(i) Definition

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Jacobi Form

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Confluent Form

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Integral Representation

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6: 4.3 Graphics

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§4.3(i) Real Arguments

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Figure 4.3.1: $\ln x$ and ${\mathrm{e}}^x$. … Magnify

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§4.3(ii) Complex Arguments: Conformal Maps

►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the negative real axis, where $w={\mathrm{e}}^z$ and $z=\ln w$ (principal value). … ►

§4.3(iii) Complex Arguments: Surfaces

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7: 11.8 Analogs to Kelvin Functions

§11.8 Analogs to Kelvin Functions

►For properties of Struve functions of argument $x{\mathrm{e}}^{\pm 3\pi \mathrm{i}/4}$ see McLachlan and Meyers (1936).

8: 4.15 Graphics

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§4.15(i) Real Arguments

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§4.15(ii) Complex Arguments: Conformal Maps

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	A	B	C	$\overline{\mathrm{C}}$	D	$\overline{\mathrm{D}}$	E	$\overline{\mathrm{E}}$	F
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Figure 4.15.7: Conformal mapping of sine and inverse sine. … Magnify

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§4.15(iii) Complex Arguments: Surfaces

… ►The corresponding surfaces for $\arccos \left(x+\mathrm{i}y\right)$, $\mathrm{arccot}\left(x+\mathrm{i}y\right)$, $\mathrm{arcsec}\left(x+\mathrm{i}y\right)$ can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).

9: 35.9 Applications

§35.9 Applications

►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument ${}_p{}^{}F_q^{}$, with $p\le 2$ and $q\le 1$. See James (1964), Muirhead (1982), Takemura (1984), Farrell (1985), and Chikuse (2003) for extensive treatments. … ►These references all use results related to the integral formulas (35.4.7) and (35.5.8). … ►In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. …

10: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

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13.19.1 $$M_{\kappa ,\mu }\left(x\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}x}{x}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{x}^{-s},$$ $\mu -\kappa \ne -\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $s$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/13.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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13.19.2 $$M_{\kappa ,\mu }\left(z\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{z}^{-s}+\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\mathrm{e}}^{-\frac{1}{2}z\pm (\frac{1}{2}+\mu -\kappa )\pi \mathrm{i}}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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13.19.3 $$W_{\kappa ,\mu }\left(z\right)\sim {\mathrm{e}}^{-\frac{1}{2}z}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

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