Bromwich integral
|
|
In mathematics, the Bromwich integral or inverse Laplace transform of F(s) is the function f(t) which has the property
- <math>\left\{\mathcal{L}f\right\}(s) = 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
External links
- Tables of Integral Transforms (http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm) at EqWorld: The World of Mathematical Equations.
Bibliography
- A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998.nl:Inverse Laplace transformatie