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

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

where <math>\mathcal{L}</math> 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:

<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).

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

See also

• Inverse Fourier transform

External links

• Tables of Integral Transforms at EqWorld: The World of Mathematical Equations.

Bibliography

• A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4nl:Inverse laplacetransformatie