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The [[Integral transform|integral transform]]

$$ 
F( x)  =  \int\limits _ { 0 } ^  \infty   e  ^ {-xt/2} ( xt) ^ {- \mu - 1/2 } W _ {\mu +
1/2, \nu }  ( xt) f( t)  dt,
$$

where  $  W _ {\mu , \nu }  ( x) $
is the Whittaker function (cf. [[Whittaker functions]]). The corresponding inversion formula is

$$ 
f( t)  =  \lim\limits _ {\lambda \rightarrow + \infty }  
\frac{1}{2 \pi i }
 
\frac{\Gamma ( 1- \mu + \nu ) }{\Gamma ( 1+ 2 \nu ) }
 \times
$$

$$ 
\times 
\int\limits _ {\beta - i \lambda } ^ {  \beta   + i \lambda } e^ {xt/2} ( xt) ^ {\mu - 1/2 } W _ {\mu - 1/2, \nu }  ( xt) F( x)  dx.
$$

For  $  \mu = \pm  \nu $
the Meijer transform becomes the [[Laplace transform]]; for  $  \mu = - 1/2 $
it becomes the  $  K _  \nu  $-
transform

$$ 
F( x)  =  
\frac{1}{\sqrt \pi }
 \int\limits _ { 0 } ^  \infty   e  ^ {-} xt/2 ( xt)  ^ {1/2} K _  \nu  \left ( 
\frac{xt}{2}
 \right ) f( t)  dt,
$$

where  $  K _  \nu  ( x) $
is the [[Macdonald function]].

The Varma transform

$$ 
F( x)  =  \int\limits _ { 0 } ^  \infty   ( xt) ^ {\nu - 1/2 } e  ^ {-} xt/2 W _ {\mu , \nu } 
( xt) f( t)  dt
$$

reduces to a Meijer transform.

The Meijer  $  K $-
transform (or the Meijer–Bessel transform) is the integral transform

$$ 
F( x)  =  \sqrt {
\frac{2} \pi 
 } \int\limits _ { 0 } ^  \infty   K _  \nu  ( xt) \sqrt xt f( t)  dt.
$$

If the function  $  f $
is locally integrable on  $  ( 0, \infty ) $, 
has bounded variation in a neighbourhood of the point  $  t = t _ {0} > 0 $, 
and if the integral

$$ 
\int\limits _ { 0 } ^  \infty   e ^ {- \beta t } | f( t) |  dt,\ \ 
\beta > \alpha \geq  0,
$$

converges, then the following inversion formula is valid:

$$ 

\frac{f( t _ {0} + 0) + f( t _ {0} - 0) }{2\ }
 =
$$

$$ 
= \ 
\lim\limits _ {\lambda \rightarrow \infty }  
\frac{1}{i \sqrt {2 \pi } }
 \int\limits _ {\beta - i \lambda
 } ^ {  \beta   + i \lambda } I _  \nu  ( t _ {0} x)( t _ {0} x)  ^ {1/2} F( x)  dx.
$$

For  $  \nu = \pm  1/2 $
the Meijer  $  K $-
transform turns into the Laplace transform.

The Meijer transform and Meijer  $  K $-
transform were introduced by C.S. Meijer in [[#References|[1]]] and, respectively, .

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.S. Meijer,   "Eine neue Erweiterung der Laplace Transformation I"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''44'''  (1941)  pp. 727–737</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  C.S. Meijer,   "Ueber eine neue Erweiterung der Laplace Transformation I"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''43'''  (1940)  pp. 599–608</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  C.S. Meijer,   "Ueber eine neue Erweiterung der Laplace Transformation II"  ''Proc. Koninkl. Ned. Akad. Wet.'' , '''43'''  (1940)  pp. 702–711</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  Y.A. Brychkov,   A.P. Prudnikov,   "Integral transforms of generalized functions" , Gordon &amp; Breach  (1989)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Ditkin,   A.P. Prudnikov,   "Operational calculus"  ''Progress in Math.'' , '''1'''  (1968)  pp. 1–75  ''Itogi Nauk. Mat. Anal. 1966''  (1967)  pp. 7–82</TD></TR></table>