Arnold's Trivium problem 51 > Calculate $$ f(k) = \int_{-\infty}^{+\infty} e^{ikx}\frac{1 - e^x}{1+e^x}dx.$$ As far as I know, this is not a function but rather the Fourier transform in tempered distributions. > 1) What is a **rigorous** proof of Arnold's problem ? > > 2) How people usually use and manipulate Fourier Transform of tempered distributions ? Edit : Thx a lot to joriki and sos440 for two very nice and rigorous proofs. To summarize, **joriki** 's idea is to get a regular integral by derivation and then using the Rediues Theorem ; while **sos440** is removing the 'constant' term that makes the integral diverge and treat it seperatly. If anyone has another different method, I'd be glad.

The poles are at $x=(2n+1)\pi\mathrm i$, and the corresponding residues are $-2\mathrm e^{-(2n+1)\pi k}$. If we ignore issues of convergence, this suggests that the integral is

$$ -4\pi\mathrm i\sum_{n=0}^\infty\mathrm e^{-(2n+1)\pi k}=\frac{-4\pi\mathrm i\mathrm e^{-\pi k}}{1-\mathrm e^{-2\pi k}}=\frac{-2\pi\mathrm i}{\sinh\pi k}\;. $$

To make this rigorous in the context of tempered distributions, you can take the derivative of the function being Fourier-transformed, calculate the resulting convergent integral, and divide by $-\mathrm ik$ (see Tempered distributions and Fourier transform at Wikipedia). The residues are slightly more complicated to evaluate, but as one might expect the result is the same.