# Fourier Transform¶

The Fourier transform is:

To show that it works:

# Laplace Transform¶

Laplace transform of is:

The contour integration is over the vertical line and is chosen large enough so that all residues are to the left of the line (that’s because the Laplace transform is only defined for larger than the residues, so we have to integrate in this range as well). It can be shown that the integral over the left semicircle goes to zero:

so the complex integral is equal to the sum of all residues of in the complex plane.

To show that it works:

where we used:

and it can be derived from the Fourier transform by transforming a function :

and making a substitution :

Where the bar () means the Laplace transform and tilde () means the Fourier transform.