We define the Meijer G function by the inverse Laplace transform

where is one of three types of integration paths , , and . 

A schematic plot of the integration path (, , or ) and the poles of the integrand () is shown below. 

Contour is one of three types of integration paths , , and .  Contour starts at and finishes at .  Contour starts at and finishes at .  Contour starts at and finishes at .  All the paths , , and put all poles on the right and all other poles of the integrand (which must be of the form ) on the left.  Define , , and to be the functions defined by the , , and contours. 

Related to this definition of Meijer G, we also define quantities , , , , , , and by , , , , , , and

Analysis of the absolute convergence of the contour integral using Stirling's asymptotic formula for the gamma function produces:

Theorem. converges absolutely if
(1) or
(2) and or
(3) , , and

Theorem. converges absolutely if
(1) or
(2) and or
(3) , , and

Theorem. converges absolutely if
(1) or
(2) and