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I have an integral on the form

$\int_{-\infty}^{\infty} e^{-k \omega' |\tau|} e^{i \tau(\omega'-\omega)} d\tau$

that I would like to simplify (or basically solve). This indeed comes from a problem involving Fourier tranforms. If $k=0$, then the solution is easy and given by the Delta function:

$\int_{-\infty}^{\infty} e^{i \tau(\omega'-\omega)} d\tau=\delta(\omega'-\omega) 2\pi$

However, for the general case of non-zero $k$, is there any way to find a nice solution?

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$$\int_{-\infty}^{\infty} e^{-k \omega' |\tau|} e^{i \tau(\omega'-\omega)} d\tau=\frac{2 k \omega'}{\left(k^2+1\right) \omega'^2+\omega^2-2 \omega\omega'},$$ for $k\omega'>0$.

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  • $\begingroup$ Much appreciated, dare I ask how you were able find this solution? $\endgroup$
    – owp
    Commented Sep 14, 2023 at 18:26
  • $\begingroup$ Ah I see, it's related to the Fourier transform of $e^{-k|x|}$, just with an added frequency shift. $\endgroup$
    – owp
    Commented Sep 14, 2023 at 18:44

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