## Monday Math 113

The Laplace Transform
Part 14: Gaussians and Error Function

Now, let’s look at the Laplace transform of a Gaussian function, specifically the function , for t≥0. From our transform definition,

We proceed by completing the square:

To simplify from here, we avoid complications of trajectory in the complex plane by assuming s real (the result can be extended to the region of convergence ). Then, by substituting , we have
.
In terms of the error function , and using , we see
,
and so
.

We can also use this to find the transform of the error function, via our integration rule:

(also with ).