The modified 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 $\mathrm{\infty }+\mathrm{I}\mathrm{\&phi;2}$ ($\mathrm{\&phi;1}<\mathrm{\&phi;2}$).

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

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 definition of the Meijer G function is related to the modified definition by

Note: See Prudnikov, Brychkov, and Marichev.

Three noticeable differences between the notations are:

the parameters of the modified Meijer G function are separated out into four natural groups,

${\&ExponentialE;}^{yz}$ instead of $z^y$ is placed inside the integral definition of ModifiedMeijerG, and

the pq\mn subscripts and superscripts which are now redundant are omitted.

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

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

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

$\mathrm{-1.205734962}\times {10}^{\mathrm{-20}}-0.\mathrm{I}$

$s≔2\left(\mathrm{sum}\left({\left(-1\right)}^i\mathrm{ModifiedMeijerG}\left(\left[\right]\&comma;\left[\right]\&comma;\left[0\right]\&comma;\left[\right]\&comma;\ln \left(z\right)+\ln \left(1+2\mathrm{I}\right)\right)\&comma;i=0..\mathrm{\infty }\right)\right)$

$s≔2\left(\sum _{i=0}^{\mathrm{\infty }}{\left(\mathrm{-1}\right)}^i\mathrm{ModifiedMeijerG}\left(\left[\right]\&comma;\left[\right]\&comma;\left[0\right]\&comma;\left[\right]\&comma;\ln \left(z\right)+\ln \left(1+2\mathrm{I}\right)\right)\right)$

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

$2\left(\sum _{i=0}^{\mathrm{\infty }}{\left(\mathrm{-1}\right)}^i{\&ExponentialE;}^{\left(\mathrm{-1}-2\mathrm{I}\right)z}\right)$

$\mathrm{convert}\left(\exp \left(z\right)\&comma;'\mathrm{ModifiedMeijerG}'\&comma;z\right)$

$\mathrm{ModifiedMeijerG}\left(\left[\right]\&comma;\left[\right]\&comma;\left[0\right]\&comma;\left[\right]\&comma;\ln \left(z\right)+\mathrm{I}\mathrm{\pi }\right)$

$\mathrm{convert}\left(\sin \left(z\right)\&comma;'\mathrm{ModifiedMeijerG}'\&comma;z\right)$

$\sqrt{\mathrm{\pi }}\mathrm{ModifiedMeijerG}\left(\left[\right]\&comma;\left[\right]\&comma;\left[\frac{1}{2}\right]\&comma;\left[0\right]\&comma;2\ln \left(z\right)-2\ln \left(2\right)\right)$

$\mathrm{convert}\left(\cos \left(z\right)\&comma;'\mathrm{ModifiedMeijerG}'\&comma;z\right)$

$\sqrt{\mathrm{\pi }}\mathrm{ModifiedMeijerG}\left(\left[\right]\&comma;\left[\right]\&comma;\left[0\right]\&comma;\left[\frac{1}{2}\right]\&comma;2\ln \left(z\right)-2\ln \left(2\right)\right)$

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

$-\mathrm{ModifiedMeijerG}\left(\left[\right]\&comma;\left[1\right]\&comma;\left[0\&comma;0\right]\&comma;\left[\right]\&comma;\ln \left(z\right)+\mathrm{I}\mathrm{\pi }\right)$

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