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Dirichlet integral

  1. Jul 2, 2010 #1
    I was trying to understand the proof given in this wiki page:


    But I'm sure their proof is correct because I don't know if the step where he says the sine is the imaginary part of the the exponential and then expands it to the whole of the integral is correct.

    [tex]\int_0^{\infty} e^{-\alpha \omega} Im( e^{i \beta \omega}) = Im( \int_0^{\infty} e^{-\alpha \omega} e^{i \beta \omega})[/tex]

    I doesn't make much sense to me. That whole part of the demonstration is very unclear to me.

    Anyone know this?
    Last edited: Jul 2, 2010
  2. jcsd
  3. Jul 2, 2010 #2
    [tex]e^{-\alpha\omega}[/tex] is real there, and since [tex]c \Im \left{a+bi\right}=\Im\left{ac+cbi\right}=cb[/tex], the equality [tex]e^{-\alpha\omega}\sin{\beta\omega}=\Im\left{e^{-\alpha\omega}e^{i\beta\omega}\right}[/tex] is valid. If you are still uncertain of this, the integral can be solved by using twice integration by parts.
  4. Jul 2, 2010 #3
    So alpha and omega have to be real for this to work.
  5. Jul 2, 2010 #4

    Gib Z

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    Homework Helper

    Yes. Also, don't forget your differentials.
  6. Jul 2, 2010 #5
    Yeah, well they do not state that alpha is real in that page. Thanks guys.
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