Meijer transform

The 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 [1] and, respectively, .

References

[1]	C.S. Meijer, "Eine neue Erweiterung der Laplace Transformation I" Proc. Koninkl. Ned. Akad. Wet. , 44 (1941) pp. 727–737
[2a]	C.S. Meijer, "Ueber eine neue Erweiterung der Laplace Transformation I" Proc. Koninkl. Ned. Akad. Wet. , 43 (1940) pp. 599–608
[2b]	C.S. Meijer, "Ueber eine neue Erweiterung der Laplace Transformation II" Proc. Koninkl. Ned. Akad. Wet. , 43 (1940) pp. 702–711
[3]	Y.A. Brychkov, A.P. Prudnikov, "Integral transforms of generalized functions" , Gordon & Breach (1989) (Translated from Russian)
[4]	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