# Mellin’s inverse formula

It may be proven, that if a function $F(s)$ has the inverse Laplace transform $f(t)$, i.e. a piecewise continuous and exponentially real function $f$ satisfying the condition

$$\mathcal{L}\{f(t)\}=F(s),$$ |

then $f(t)$ is uniquely determined when not regarded as different such functions which differ from each other only in a point set having Lebesgue measure zero.

The inverse Laplace transform is directly given by Mellin’s inverse formula

$$f(t)=\frac{1}{2\pi i}{\int}_{\gamma -i\mathrm{\infty}}^{\gamma +i\mathrm{\infty}}{e}^{st}F(s)\mathit{d}s,$$ |

by the Finn R. H. Mellin (1854—1933). Here it must be integrated along a straight line parallel^{} to the imaginary axis^{} and intersecting the real axis in the point $\gamma $ which must be chosen so that it is greater than the real parts of all singularities of $F(s)$.

In practice, computing the complex integral can be done by using the Cauchy residue theorem.

Title | Mellin’s inverse formula |

Canonical name | MellinsInverseFormula |

Date of creation | 2013-03-22 14:23:02 |

Last modified on | 2013-03-22 14:23:02 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 13 |

Author | pahio (2872) |

Entry type | Result |

Classification | msc 44A10 |

Synonym | inverse Laplace transformation |

Synonym | Bromwich integral |

Synonym | Fourier-Mellin integral |

Related topic | InverseLaplaceTransformOfDerivatives |

Related topic | HjalmarMellin |

Related topic | TelegraphEquation |