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I've been trying to solve this problem, but I don't know how to begin solving it. I know that the residue theorem must be used at some point, but not much else. Any help is welcome.

Let be the parallelogram $PR$, defined by the points $R+Ri$, $R+1+Ri$, $−R−RI$ and $−R + 1 − Ri$. Integrating the function $f(z) = e^{iπz^2} tan(πz)$ over $PR$ and tending $R \rightarrow \infty$ , prove that:

$$ \int_{- \infty}^{\infty} {e^{-x^2}dx}= \sqrt{\pi}$$

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    $\begingroup$ Hi, welcome to Math SE. It might be better to list the vertices in the order $R+Ri,\,R+1+Ri,\,-R+1-Ri,\,-R-Ri$. As for your problem, you want to do something similar to the tenth proof here. $\endgroup$ Commented 3 hours ago
  • $\begingroup$ @J.G. I'll edit it, that's how the problem was written in my assigment. Thank you so much for the resource! $\endgroup$ Commented 3 hours ago

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