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单词 MellinsInverseFormula
释义

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

{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)=12πiγ-iγ+iestF(s)𝑑s,

by the Finn R. H. Mellin (1854—1933).  Here it must be integrated along a straight line parallelMathworldPlanetmathPlanetmath to the imaginary axisMathworldPlanetmath and intersecting the real axis in the point γ 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.

TitleMellin’s inverse formula
Canonical nameMellinsInverseFormula
Date of creation2013-03-22 14:23:02
Last modified on2013-03-22 14:23:02
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id13
Authorpahio (2872)
Entry typeResult
Classificationmsc 44A10
Synonyminverse Laplace transformation
SynonymBromwich integral
SynonymFourier-Mellin integral
Related topicInverseLaplaceTransformOfDerivatives
Related topicHjalmarMellin
Related topicTelegraphEquation
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