Thread: Integrals involving complex exponentials

Hello Everyone!

I'm being reluctant when finding integrals involving complex exponentials, of the form:

1.
2.

This is because I do not know what to do with infinities multiplied by a complex number, but I know as

Is there a way to solve this without getting into Fourier transform? I know that generalized functions are involved in those integrals, but can we got an answer without going through that?

Thanks!

Those integrals do not converge. Period.

Originally Posted by Ackbeet

Those integrals do not converge. Period.

Okay. So let's put in place of , we get kinda reminds me of something

Isn't that the Fourier transform of ?

Originally Posted by rebghb

Okay. So let's put in place of , we get kinda reminds me of something

Isn't that the Fourier transform of ?

Whatever some people think that is not how the FT of f(x)=1 is defined (at least without specifying what convergence method you want to use)

CB

I know this integral, eventually gives . But that's like giving infity a function in other variable

Originally Posted by rebghb

I know this integral, eventually gives . But that's like giving infity a function in other variable

Not under a conventional definition of integration. You need to resort to the theory of distributions to get the delta functional.

CB

Alright, before closing the thread, I would like to say, In a course on basic signal proecssing, they establish the following:



Now, by inverse transform, and exchanging variables, we get the that the Fourier transform of f(t)=1 is . Is this mathematically accepted?

NB The delta here is the Dirac delta

Originally Posted by rebghb

Alright, before closing the thread, I would like to say, In a course on basic signal proecssing, they establish the following:



Now, by inverse transform, and exchanging variables, we get the that the Fourier transform of f(t)=1 is . Is this mathematically accepted?

NB The delta here is the Dirac delta

Yes, in a slightly hand-waving way, but acceptable in signal processing and physics

CB