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Math Help - Integrals involving complex exponentials

  1. #1
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    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!
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  2. #2
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    Re: Integrals involving complex exponentials

    Those integrals do not converge. Period.
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  3. #3
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    Re: Integrals involving complex exponentials

    Quote Originally Posted by Ackbeet View Post
    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

    Isn't that the Fourier transform of f(t) = 1?
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    Re: Integrals involving complex exponentials

    Quote Originally Posted by rebghb View Post
    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

    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
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  5. #5
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    Re: Integrals involving complex exponentials

    I know this integral, eventually gives \delta (f). But that's like giving infity a function in other variable
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  6. #6
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    Re: Integrals involving complex exponentials

    Quote Originally Posted by rebghb View Post
    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
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  7. #7
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    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
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  8. #8
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    Re: Integrals involving complex exponentials

    Quote Originally Posted by rebghb View Post
    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
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