# Meijer transform

where is the Whittaker function (cf. Whittaker functions). The corresponding inversion formula is

For the Meijer transform becomes the Laplace transform; for it becomes the -transform

where is the Macdonald function.

The Varma transform

reduces to a Meijer transform.

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

If the function is locally integrable on , has bounded variation in a neighbourhood of the point , and if the integral

converges, then the following inversion formula is valid:

For the Meijer -transform turns into the Laplace transform.

The Meijer transform and Meijer -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 |

**How to Cite This Entry:**

Meijer transform. Yu.A. BrychkovA.P. Prudnikov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Meijer_transform&oldid=12567