Fourier transform - application, integral calculation

I'm trying to brush the dust of my transform tinkering. (Have to use it in some electronic stuff.) I just did an exercise where I calculate the F-transform of an even function.

Let f(x) be:

Find the even Fourier transform.

I calculated the transform with this formula

Knowing that the function is even I write:

A bit integration gives, don't hesitate to call out errors.

Now as an application to this calculation I'm supposed to determine this integral.

This is where I like to get some help... I simply don't know how to do it. Calculating the integral, in an other way, I get

Re: Fourier transform - application, integral calculation

Quote:

Originally Posted by

**liquidFuzz** I'm trying to brush the dust of my transform tinkering. (Have to use it in some electronic stuff.) I just did an exercise where I calculate the F-transform of an even function.

Let f(x) be:

Find the even Fourier transform.

I calculated the transform with this formula

Knowing that the function is even I write:

A bit integration gives, don't hesitate to call out errors.

Now as an application to this calculation I'm supposed to determine this integral.

This is where I like to get some help... I simply don't know how to do it. Calculating the integral, in an other way, I get

The notation for the Fourier transform of a function should be different from the notation for the function itself. So you should write rather than

The Fourier inversion theorem says that you can retrieve from by the formula Put t=0 in that formula and you will get the result you are looking for.

Re: Fourier transform - application, integral calculation

Thanks! For slapping my fingers, and explaining how to do this.

Re: Fourier transform - application, integral calculation

How about this..?

Assume that converges to f(x), , and that the Fourier transform is correct,

Taking a arbitrary point on f(x)

Gives