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11: 35.3 Multivariate Gamma and Beta Functions

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35.3.2 $${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{s_m-\frac{1}{2}(m+1)}\prod _{j=1}^{m-1}{|{(𝐗)}_j|}^{s_j-s_{j+1}}d𝐗,$$ $s_j\in ℂ$, $\mathrm{\Re }\left(s_j\right)>\frac{1}{2}(j-1)$, $j=1,\mathrm{\dots },m$.

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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12: 9.17 Methods of Computation

… ►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. …

13: 22.10 Maclaurin Series

… ►The radius of convergence is the distance to the origin from the nearest pole in the complex $k$-plane in the case of (22.10.4)–(22.10.6), or complex $k^{\prime }$-plane in the case of (22.10.7)–(22.10.9); see §22.17. …

14: 19.21 Connection Formulas

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19.21.1 $$R_F(0,z+1,z)R_D(0,z+1,1)+R_D(0,z+1,z)R_F(0,z+1,1)=3\pi /(2z),$$ $z\in ℂ\setminus (-\mathrm{\infty },0]$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $\in$: element of, $(a,b]$: half-closed interval and $\setminus$: set subtraction

Referenced by:

§19.21(i), §19.21(i), §19.7(i)

Permalink:

http://dlmf.nist.gov/19.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.21(i), §19.21 and Ch.19

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15: 8.13 Zeros

… ►For information on the distribution and computation of zeros of $\gamma (a,\lambda a)$ and $\mathrm{\Gamma }(a,\lambda a)$ in the complex $\lambda$-plane for large values of the positive real parameter $a$ see Temme (1995a). … ►When $x>x_n^{\ast }$ a pair of conjugate trajectories emanate from the point $a=a_n^{\ast }$ in the complex $a$-plane. …

16: 4.13 Lambert $W$-Function

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4.13.5_1 $${\left(\frac{W_0\left(z\right)}{z}\right)}^a={\mathrm{e}}^{-a{W}_0\left(z\right)}=\sum _{n=0}^{\mathrm{\infty }}\frac{a{\left(n+a\right)}^{n-1}}{n!}{\left(-z\right)}^n,$$ , $a\in ℂ$.

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Symbols:

$W\left(z\right)$: Lambert $W$-function, $ℂ$: complex plane, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $n$: integer, $a$: real or complex constant and $z$: complex variable

Notes:

See Corless et al. (1996, formula (2.36)).

Referenced by:

§4.13, Erratum (V1.1.7) for Expansion

Permalink:

http://dlmf.nist.gov/4.13.E5_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

See also:

Annotations for §4.13 and Ch.4

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17: 19.26 Addition Theorems

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19.26.17 $$\sqrt{\alpha }{R}_C(\beta ,\alpha +\beta )+\sqrt{\beta }{R}_C(\alpha ,\alpha +\beta )=\pi /2,$$ $\alpha ,\beta \in ℂ\setminus (-\mathrm{\infty },0)$, $\alpha +\beta >0$.

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $\in$: element of, $(a,b)$: open interval, $\setminus$: set subtraction and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.26(ii)

Permalink:

http://dlmf.nist.gov/19.26.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §19.26(ii), §19.26 and Ch.19

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18: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.2 $$P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)=\frac{{\mathrm{\Gamma }}_m\left(\gamma +\nu +\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(\gamma +\frac{1}{2}(m+1)\right)}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\nu ,\gamma +\delta +\nu +\frac{1}{2}(m+1)}{\gamma +\frac{1}{2}(m+1)};𝐓),$$ ; $\gamma ,\delta ,\nu \in ℂ$; $\mathrm{\Re }\left(\gamma \right)>-1$.

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Defines:

$P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$: Jacobi function of matrix argument

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $ℂ$: complex plane, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ 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

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19: Bibliography G

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A continued fraction algorithm for the computation of higher transcendental functions in the complex plane. Math. Comp. 21 (97), pp. 18–29.

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External Links:

FullText, MathReview (D. C. Gilles), ZentralBlatt

Referenced by:

§10.74(v), §3.10(ii).

Encodings:

BibTeX

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Painlevé Differential Equations in the Complex Plane. Studies in Mathematics, Vol. 28, Walter de Gruyter & Co., Berlin-New York.

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External Links:

ISBN 3-11-017379-4, MathReview, ZentralBlatt

Referenced by:

§32.10(i), §32.10(iii), §32.10(iv), §32.10(v), §32.10(vi), §32.2(iv), §32.7(iii), §32.7(iv), §32.7(v), §32.7(vi), §32.7(vii), §32.8(ii), §32.8(iii), §32.8(iv), §32.8(v), §32.9(i), §32.9(ii), §32.9(iii).

Encodings:

BibTeX

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20: Bibliography O

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General connection formulae for Liouville-Green approximations in the complex plane. Philos. Trans. Roy. Soc. London Ser. A 289, pp. 501–548.

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External Links:

ISSN 0080-4614, FullText, MathReview (W. Wasow), ZentralBlatt

Referenced by:

§2.8(v), (9.13.18), §9.13(i), §9.13(i).

Encodings:

BibTeX

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