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
