# Integrals involving complex exponentials

• Sep 28th 2011, 09:01 AM
rebghb
Integrals involving complex exponentials
Hello Everyone!

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

1. $\int^{+\infty}_{-\infty}e^{-iat}dt$
2. $\int^{+\infty}_{0}e^{-iat}dt$

This is because I do not know what to do with infinities multiplied by a complex number, but I know as $t \rightarrow \infty$

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!
• Sep 28th 2011, 10:35 AM
Ackbeet
Re: Integrals involving complex exponentials
Those integrals do not converge. Period.
• Sep 29th 2011, 09:58 AM
rebghb
Re: Integrals involving complex exponentials
Quote:

Originally Posted by Ackbeet
Those integrals do not converge. Period.

Okay. So let's put $2\pi f$ in place of $a$, we get $\int ^{+\infty}_{-\infty}e^{-i2\pi ft}dt$ kinda reminds me of something (Happy)

Isn't that the Fourier transform of $f(t) = 1$?
• Sep 29th 2011, 07:14 PM
CaptainBlack
Re: Integrals involving complex exponentials
Quote:

Originally Posted by rebghb
Okay. So let's put $2\pi f$ in place of $a$, we get $\int ^{+\infty}_{-\infty}e^{-i2\pi ft}dt$ kinda reminds me of something (Happy)

Isn't that the Fourier transform of $f(t) = 1$?

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
• Sep 30th 2011, 11:26 AM
rebghb
Re: Integrals involving complex exponentials
I know this integral, eventually gives $\delta (f)$. But that's like giving infity a function in other variable
• Sep 30th 2011, 12:12 PM
CaptainBlack
Re: Integrals involving complex exponentials
Quote:

Originally Posted by rebghb
I know this integral, eventually gives $\delta (f)$. 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
• Oct 1st 2011, 01:44 AM
rebghb
Re: Integrals involving complex exponentials
Alright, before closing the thread, I would like to say, In a course on basic signal proecssing, they establish the following:

$\mathcal{F} {\delta (t)} =\int ^{+\infty}_{-\infty}\delta (t) e^{-i2\pi ft dt} = 1$

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

NB The delta here is the Dirac delta
• Oct 1st 2011, 06:24 AM
CaptainBlack
Re: Integrals involving complex exponentials
Quote:

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:

$\mathcal{F} {\delta (t)} =\int ^{+\infty}_{-\infty}\delta (t) e^{-i2\pi ft dt} = 1$

Now, by inverse transform, and exchanging variables, we get the that the Fourier transform of f(t)=1 is $\delta (f)$. 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