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!

Re: Integrals involving complex exponentials

Those integrals do not converge. Period.

Re: Integrals involving complex exponentials

Quote:

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 (Happy)

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

Re: Integrals involving complex exponentials

Quote:

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 (Happy)

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

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

Re: Integrals involving complex exponentials

Quote:

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

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

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:

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

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