**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 ).

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Tags: Error Function, Gaussian, Laplace Transform, Math, Monday Math

This entry was posted on March 22, 2010 at 2:04 pm and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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