The Meijer G function is defined by the inverse Laplace transform

where

and  L is one of three types of integration paths $L_{\mathrm{\gamma }+\mathrm{\infty }\mathrm{I}}$, $L_{\mathrm{\infty }}$, and $L_{-\mathrm{\infty }}$.

Contour $L_{\mathrm{\infty }}$ starts at $\mathrm{\infty }+\mathrm{I}\mathrm{\&phi;1}$ and finishes at $\mathrm{\infty }+\mathrm{I}\mathrm{\&phi;2}\left(\mathrm{\&phi;1}<\mathrm{\&phi;2}\right)$.

Contour $L_{-\mathrm{\infty }}$ starts at $-\mathrm{\infty }+\mathrm{I}\mathrm{\&phi;1}$ and finishes at $-\mathrm{\infty }+\mathrm{I}\mathrm{\&phi;2}\left(\mathrm{\&phi;1}<\mathrm{\&phi;2}\right)$.

Contour $L_{\mathrm{\gamma }+\mathrm{\infty }\mathrm{I}}$ starts at $\mathrm{\gamma }-\mathrm{\infty }$ and finishes at $\mathrm{\gamma }+\mathrm{\infty }\mathrm{I}$.

All the paths $L_{\mathrm{\infty }}$, $L_{-\mathrm{\infty }}$, and $L_{\mathrm{\gamma }+\mathrm{\infty }\mathrm{I}}$ 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

Note: See Prudnikov, Brychkov, and Marichev.

The MeijerG function satisfies the following $q$th-order linear differential equation

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

$\mathrm{MeijerG}\left(\left[\left[1\&comma;1\&comma;1\&comma;1\right]\&comma;\left[\right]\right]\&comma;\left[\left[\right]\&comma;\left[4\&comma;3\&comma;2\&comma;2\right]\right]\&comma;\mathrm{\pi }\right)$

$\mathrm{MeijerG}\left(\left[\left[1\&comma;1\&comma;1\&comma;1\right]\&comma;\left[\right]\right]\&comma;\left[\left[\right]\&comma;\left[4\&comma;3\&comma;2\&comma;2\right]\right]\&comma;\mathrm{\pi }\right)$

$\mathrm{evalf}\left(\right)$

$8.898308178\times {10}^{\mathrm{-28}}+9.796677125\times {10}^{\mathrm{-26}}\mathrm{I}$

$s≔\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[\right]\right]\&comma;\left[\left[0\right]\&comma;\left[\right]\right]\&comma;z\left(1+2\mathrm{I}\right)\right)$

$s≔\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[\right]\right]\&comma;\left[\left[0\right]\&comma;\left[\right]\right]\&comma;\left(1+2\mathrm{I}\right)z\right)$

$\mathrm{convert}\left(s\&comma;'\mathrm{StandardFunctions}'\right)$

${\&ExponentialE;}^{\left(\mathrm{-1}-2\mathrm{I}\right)z}$

$\mathrm{convert}\left(\exp \left(z\right)\&comma;'\mathrm{MeijerG}'\&comma;\mathrm{include}=\mathrm{elementary}\right)$

$\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[\right]\right]\&comma;\left[\left[0\right]\&comma;\left[\right]\right]\&comma;-z\right)$

$\mathrm{convert}\left(\sin \left(z\right)\&comma;'\mathrm{MeijerG}'\&comma;\mathrm{include}=\mathrm{elementary}\right)$

$\sqrt{\mathrm{\pi }}\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[\right]\right]\&comma;\left[\left[\frac{1}{2}\right]\&comma;\left[0\right]\right]\&comma;\frac{z^2}{4}\right)$

$\mathrm{convert}\left(\cos \left(z\right)\&comma;'\mathrm{MeijerG}'\&comma;\mathrm{include}=\mathrm{elementary}\right)$

$\sqrt{\mathrm{\pi }}\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[\right]\right]\&comma;\left[\left[0\right]\&comma;\left[\frac{1}{2}\right]\right]\&comma;\frac{z^2}{4}\right)$

$\mathrm{convert}\left(\mathrm{Ei}\left(z\right)\&comma;'\mathrm{MeijerG}'\right)$

$-\mathrm{MeijerG}\left(\left[\left[\right]\&comma;\left[1\right]\right]\&comma;\left[\left[0\&comma;0\right]\&comma;\left[\right]\right]\&comma;-z\right)$

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