Monday Math 118

The Laplace Transform
Part 19: Inverse Laplace Transform

Recall from here the Fourier transform
and its inverse
Combining these, we see
Now, letting f(t)=0 for t<0, the above becomes

Now, suppose this f(t) has a Laplace transform F(s) with region of convergence \Re(s)>\sigma. Thus, for a real number c>σ, the line \Re(s)=c lies entirely in the region of convergence. So let us make in the change of variables s=c+\imath\omega in the above integral; then ds=\imath\,d\omega, so d\omega=\frac{ds}{\imath}, 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 \Re(s)=c. This is known as the Bromwich integral. However, actually evaluating this integral is generally very difficult.


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