# Thread: Integrals involving complex exponentials

1. ## Integrals involving complex exponentials

Hello Everyone!

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

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

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

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!

2. ## Re: Integrals involving complex exponentials

Those integrals do not converge. Period.

3. ## Re: Integrals involving complex exponentials

Originally Posted by Ackbeet
Those integrals do not converge. Period.
Okay. So let's put $\displaystyle 2\pi f$ in place of$\displaystyle a$, we get $\displaystyle \int ^{+\infty}_{-\infty}e^{-i2\pi ft}dt$ kinda reminds me of something

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

4. ## Re: Integrals involving complex exponentials

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

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

5. ## Re: Integrals involving complex exponentials

I know this integral, eventually gives $\displaystyle \delta (f)$. But that's like giving infity a function in other variable

6. ## Re: Integrals involving complex exponentials

Originally Posted by rebghb
I know this integral, eventually gives $\displaystyle \delta (f)$. 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

7. ## 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:

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

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

NB The delta here is the Dirac delta

8. ## Re: Integrals involving complex exponentials

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:

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

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