The objective is to find the $n$-th Fourier transform of function $e^{-|x|}$. So i started of with finding the first Fourier transform and the result is $\frac{2}{y^2+1}$. Now I wanted to find its Fourier transform:
$$\int_{\mathbb R} \frac{2}{x^2+1}e^{-ixy} \, dx=\int_{C}\frac{2}{z^2+1}e^{-izy} \, dx-\int_{\Gamma}\frac{2}{z^2+1}e^{-izy} \, dz,$$ where $\Gamma$ is the upper half of a circle or radius r, and $C$ is $\Gamma \cup (-r,r) $.
Using residue theorem I can obtain the integral over C: $$\int_{C}\frac{2}{x^2+1}e^{-ixy}\,dx=2\pi i \frac{2}{i+i}e^{-iyi}=2\pi e^y$$ and the integral over $\Gamma$ converges to $0$ when $r\rightarrow \infty$. So the second fourier transform is $2\pi e^{y}$. Where at this point I made a mistake?