# Mellin transform

The *Mellin transform ^{}* is an integral transform

^{}defined as follows:

$$F(s)={\int}_{0}^{\mathrm{\infty}}f(t){t}^{s-1}\mathit{d}t$$ |

Intuitively, it may be viewed as a continuous^{} analogue of a power series
— instead of synthetizing a function^{} by summing multiples of integer
powers, we integrate over all real powers. This transform is closely
related to the Laplace transform^{} — if we make a change of variables
$t={e}^{-r}$ and define $g$ by $f({e}^{-r})=g(r)$, then the above integral^{}
becomes

$$F(s)=-{\int}_{-\mathrm{\infty}}^{+\mathrm{\infty}}g(r){e}^{-rs}\mathit{d}r,$$ |

which is a bilateral Laplace transform.

(more to come)

Title | Mellin transform |
---|---|

Canonical name | MellinTransform |

Date of creation | 2015-02-17 15:10:57 |

Last modified on | 2015-02-17 15:10:57 |

Owner | rspuzio (6075) |

Last modified by | pahio (2872) |

Numerical id | 7 |

Author | rspuzio (2872) |

Entry type | Definition |

Classification | msc 44A15 |