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MeijerG

Meijer G function

 Calling Sequence MeijerG([as, bs], [cs, ds], z)

Parameters

 as - list of the form [a1, ..., am]; first group of numerator $\mathrm{\Gamma }$ parameters bs - list of the form [b1, ..., bn]; first group of denominator $\mathrm{\Gamma }$ parameters cs - list of the form [c1, ..., cp]; second group of numerator $\mathrm{\Gamma }$ parameters ds - list of the form [d1, ..., dq]; second group of denominator $\mathrm{\Gamma }$ parameters z - expression

Description

 • The Meijer G function is defined by the inverse Laplace transform

$\mathrm{MeijerG}\left(\left[\mathrm{as},\mathrm{bs}\right],\left[\mathrm{cs},\mathrm{ds}\right],z\right)=\frac{1}{2\mathrm{\pi }I}\underset{L}{\oint }\frac{\Gamma \left(1-\mathrm{as}+y\right)\Gamma \left(\mathrm{cs}-y\right)}{\Gamma \left(\mathrm{bs}-y\right)\Gamma \left(1-\mathrm{ds}+y\right)}{z}^{y}ⅆy$

 where

$\mathrm{as}=\left[\mathrm{a1},\mathrm{...},\mathrm{am}\right],\mathrm{\Gamma }\left(1-\mathrm{as}+y\right)=\mathrm{\Gamma }\left(1-\mathrm{a1}+y\right)\mathrm{...}\mathrm{\Gamma }\left(1-\mathrm{am}+y\right)$

$\mathrm{bs}=\left[\mathrm{b1},\mathrm{...},\mathrm{bn}\right],\mathrm{\Gamma }\left(\mathrm{bs}-y\right)=\mathrm{\Gamma }\left(\mathrm{b1}-y\right)\mathrm{...}\mathrm{\Gamma }\left(\mathrm{bn}-y\right)$

$\mathrm{cs}=\left[\mathrm{c1},\mathrm{...},\mathrm{cp}\right],\mathrm{\Gamma }\left(\mathrm{cs}-y\right)=\mathrm{\Gamma }\left(\mathrm{c1}-y\right)\mathrm{...}\mathrm{\Gamma }\left(\mathrm{cp}-y\right)$

$\mathrm{ds}=\left[\mathrm{d1},\mathrm{...},\mathrm{dq}\right],\mathrm{\Gamma }\left(1-\mathrm{ds}+y\right)=\mathrm{\Gamma }\left(1-\mathrm{d1}+y\right)\mathrm{...}\mathrm{\Gamma }\left(1-\mathrm{dq}+y\right)$

 and  L is one of three types of integration paths ${L}_{\mathrm{\gamma }+\mathrm{\infty }I}$, ${L}_{\mathrm{\infty }}$, and ${L}_{-\mathrm{\infty }}$.
 Contour ${L}_{\mathrm{\infty }}$ starts at $\mathrm{\infty }+I\mathrm{φ1}$ and finishes at $\mathrm{\infty }+I\mathrm{φ2}\left(\mathrm{φ1}<\mathrm{φ2}\right)$.
 Contour ${L}_{-\mathrm{\infty }}$ starts at $-\mathrm{\infty }+I\mathrm{φ1}$ and finishes at $-\mathrm{\infty }+I\mathrm{φ2}\left(\mathrm{φ1}<\mathrm{φ2}\right)$.
 Contour ${L}_{\mathrm{\gamma }+\mathrm{\infty }I}$ starts at $\mathrm{\gamma }+-\mathrm{\infty }$ and finishes at $\mathrm{\gamma }+\mathrm{\infty }I$.
 All the paths ${L}_{\mathrm{\infty }}$, ${L}_{-\mathrm{\infty }}$, and ${L}_{\mathrm{\gamma }+\mathrm{\infty }I}$ put all $\mathrm{cj}+k$ poles on the right and all other poles of the integrand (which must be of the form $\mathrm{aj}-1+k$) on the left.
 • The classical notation used to represent the MeijerG function relates to the notation used in Maple by

${G}_{\mathrm{pq}}^{\mathrm{mn}}\left(z|{}_{{b}_{1},\dots ,{b}_{m},{b}_{m+1},\dots ,{b}_{q}}^{{a}_{1},\dots ,{a}_{n},{a}_{n+1},\dots ,{a}_{p}}\right)=\mathrm{MeijerG}\left(\left[\left[{a}_{1},\dots ,{a}_{n}\right],\left[{a}_{n+1},\dots ,{a}_{p}\right]\right],\left[\left[{b}_{1},\dots ,{b}_{m}\right],\left[{b}_{m+1},\dots ,{b}_{q}\right]\right],z\right)$

 Note: See Prudnikov, Brychkov, and Marichev.
 The MeijerG function satisfies the following $q$th-order linear differential equation

$\left({\left(-1\right)}^{p-m-n}x\left(\prod _{i=1}^{p}\left(x\mathrm{D}-{a}_{i}+1\right)\right)-\left(\prod _{i=1}^{q}\left(x\mathrm{D}-{b}_{i}\right)\right)\right)y\left(x\right)=0$

 where $'\mathrm{D}'=\frac{d}{\mathrm{dx}}$ and p is less than or equal to q.

Examples

 > $\mathrm{MeijerG}\left(\left[\left[1,1,1,1\right],\left[\right]\right],\left[\left[\right],\left[4,3,2,2\right]\right],\mathrm{\pi }\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[{1}{,}{1}{,}{1}{,}{1}\right]{,}\left[\right]\right]{,}\left[\left[\right]{,}\left[{4}{,}{3}{,}{2}{,}{2}\right]\right]{,}{\mathrm{\pi }}\right)$ (1)
 > $\mathrm{evalf}\left(\right)$
 ${8.898308178}{×}{{10}}^{{-28}}{+}{9.796677125}{×}{{10}}^{{-26}}{}{I}$ (2)
 > $s≔\mathrm{MeijerG}\left(\left[\left[\right],\left[\right]\right],\left[\left[0\right],\left[\right]\right],z\left(1+2I\right)\right)$
 ${s}{≔}{\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[\right]\right]{,}\left({1}{+}{2}{}{I}\right){}{z}\right)$ (3)
 > $\mathrm{convert}\left(s,'\mathrm{StandardFunctions}'\right)$
 ${{ⅇ}}^{\left({-1}{-}{2}{}{I}\right){}{z}}$ (4)
 > $\mathrm{convert}\left(\mathrm{exp}\left(z\right),'\mathrm{MeijerG}',\mathrm{include}=\mathrm{elementary}\right)$
 ${\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[\right]\right]{,}{-}{z}\right)$ (5)
 > $\mathrm{convert}\left(\mathrm{sin}\left(z\right),'\mathrm{MeijerG}',\mathrm{include}=\mathrm{elementary}\right)$
 $\sqrt{{\mathrm{\pi }}}{}{\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[\frac{{1}}{{2}}\right]{,}\left[{0}\right]\right]{,}\frac{{{z}}^{{2}}}{{4}}\right)$ (6)
 > $\mathrm{convert}\left(\mathrm{cos}\left(z\right),'\mathrm{MeijerG}',\mathrm{include}=\mathrm{elementary}\right)$
 $\sqrt{{\mathrm{\pi }}}{}{\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[\right]\right]{,}\left[\left[{0}\right]{,}\left[\frac{{1}}{{2}}\right]\right]{,}\frac{{{z}}^{{2}}}{{4}}\right)$ (7)
 > $\mathrm{convert}\left(\mathrm{Ei}\left(z\right),'\mathrm{MeijerG}'\right)$
 ${-}{\mathrm{MeijerG}}{}\left(\left[\left[\right]{,}\left[{1}\right]\right]{,}\left[\left[{0}{,}{0}\right]{,}\left[\right]\right]{,}{-}{z}\right)$ (8)

References

 Prudnikov, A. P.; Brychkov, Yu; and Marichev, O. Integrals and Series, Volume 3: More Special Functions. New York: Gordon and Breach Science Publishers, 1990.