Mellin%E2%80%93Barnes%20integrals

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21: 7.7 Integral Representations

§7.7 Integral Representations

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§7.7(i) Error Functions and Dawson’s Integral

►Integrals of the type $\int {\mathrm{e}}^{-z^2}R(z)dz$, where $R(z)$ is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions. … ►

§7.7(ii) Auxiliary Functions

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Mellin–Barnes Integrals

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22: 2.3 Integrals of a Real Variable

… ►For the Fourier integral … ►where $ℳf\left(\alpha \right)$ is the Mellin transform of $f(t)$ (§2.5(i)). … ►In the integral … ►

§2.3(vi) Asymptotics of Mellin Transforms

►For the asymptotics of the Mellin transform $ℳf\left(z\right)={\int }_0^{\mathrm{\infty }}{t}^{z-1}f(t)dt$ as $z\to \mathrm{\infty }$ see Frenzen (1987b), Sidi (1985, 2011).

23: 13.16 Integral Representations

§13.16 Integral Representations

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§13.16(i) Integrals Along the Real Line

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§13.16(ii) Contour Integrals

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§13.16(iii) Mellin–Barnes Integrals

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24: 11.5 Integral Representations

§11.5 Integral Representations

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§11.5(i) Integrals Along the Real Line

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§11.5(ii) Contour Integrals

►For loop-integral versions of (11.5.1), (11.5.2), (11.5.4), and (11.5.7) see Babister (1967, §§3.3 and 3.14). ►

Mellin–Barnes Integrals

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25: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

… ►is called an elliptic integral. … ►

§19.2(ii) Legendre’s Integrals

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§19.2(iii) Bulirsch’s Integrals

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§19.2(iv) A Related Function: $R_C(x,y)$

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26: 13.23 Integrals

§13.23 Integrals

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§13.23(i) Laplace and Mellin Transforms

… ►For additional Laplace and Mellin transforms see Erdélyi et al. (1954a, §§4.22, 5.20, 6.9, 7.5), Marichev (1983, pp. 283–287), Oberhettinger and Badii (1973, §1.17), Oberhettinger (1974, §§1.13, 2.8), and Prudnikov et al. (1992a, §§3.34, 3.35). … ► … ►

§13.23(v) Other Integrals

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27: 14.17 Integrals

§14.17 Integrals

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§14.17(i) Indefinite Integrals

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§14.17(ii) Barnes’ Integral

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§14.17(vi) Mellin Transforms

►For Mellin transforms involving associated Legendre functions see Oberhettinger (1974, pp. 69–82) and Marichev (1983, pp. 247–283), and for inverse transforms see Oberhettinger (1974, pp. 205–215).

28: 13.4 Integral Representations

§13.4 Integral Representations

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§13.4(ii) Contour Integrals

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§13.4(iii) Mellin–Barnes Integrals

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29: 13.10 Integrals

§13.10 Integrals

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§13.10(i) Indefinite Integrals

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Loop Integrals

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§13.10(iii) Mellin Transforms

… ►For additional Mellin transforms see Erdélyi et al. (1954a, §§6.9, 7.5), Marichev (1983, pp. 283–287), and Oberhettinger (1974, §§1.13, 2.8). …

30: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.12 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{\gamma -\frac{1}{2}(m+1)}{}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};-𝐗)d𝐗={\mathrm{\Gamma }}_m\left(\gamma \right){\left|𝐓\right|}^{-\gamma }{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,\gamma }{b_1,\mathrm{\dots },b_q};-{𝐓}^{-1}),$$ $\mathrm{\Re }\left(\gamma \right)>\frac{1}{2}(m-1)$.

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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, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable, $p$: nonnegative integer, $q$: nonnegative integer and $m$: positive integer

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http://dlmf.nist.gov/35.8.E12

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See also:

Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

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Euler Integral

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§35.8(v) Mellin–Barnes Integrals

►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. …These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case $m=1$. …