Bromwich integral

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

{\mathcal  {L}}\left\{f(t)\right\}=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).

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