**The Laplace Transform**

Part 19: Inverse Laplace Transform

Recall from here the Fourier transform

and its inverse

.

Combining these, we see

.

Now, letting for *t*<0, the above becomes

.

Now, suppose this *f*(*t*) has a Laplace transform *F*(*s*) with region of convergence . Thus, for a real number *c*>*σ*, the line lies entirely in the region of convergence. So let us make in the change of variables in the above integral; then , so , and we get, with some rearranging, that

,

and thus, the inverse of the Laplace transform is given by

,

where the contour of integration is over the line . This is known as the Bromwich integral. However, actually evaluating this integral is generally very difficult.

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Tags: Bromwich Integral, Contour Integration, Fourier Transform, Inverse Laplace Transform, Laplace Transform, Math, Monday Math

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