Bromwich integral

In mathematics, the Bromwich integral of F(s) is the function f(t) which has the property

<math>\mathcal{L}\left\{f(t)\right\}(s) = F(s),</math>

and is thus sometimes simply called the inverse Laplace transform.

The Laplace transform and the inverse Laplace transform together have a number of properties that make them useful for analysing linear dynamic systems.

The Bromwich integral, also called the Fourier-Mellin integral, is a path integral defined by:

<math>f(t) = \frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}F(s)e^{st}\,ds,\quad t>0,</math>

where the integration is done along the vertical line x=c in the complex plane such that c is greater than the real part of all singularities of F(s).