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# Bromwich integral

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

$\left\{\mathcal{L}f\right\}(s) = F(s),$

where $\mathcal{L}$ is the Laplace transform. The Bromwich integral 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:

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

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).

The name is for Thomas John I'Anson Bromwich (1875-1929).