Mellin's inverse formula MellinsInverseFormula 2004-05-31 15:35:28 2005-06-13 13:37:28 Result Laplace transform % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here It may be proven, that if a function $F(s)$ has the {\em inverse Laplace transform} $f(t)$, i.e. a piecewise continuous and exponentially \PMlinkescapetext{restricted} 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 {\em Mellin's inverse formula} $$f(t)= \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}e^{st}F(s)\,ds,$$ 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.