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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?

Max
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    The integral over $\Gamma$ only converges to $0$ as $r\to\infty$ if $y$ satisfies a certain condition. Knowing what you should get as the result should enable you to find the mistake and (hopefully) fix it. – Daniel Fischer Nov 15 '14 at 17:46
  • There are two poles associated with the denominator x^2+1, namely +i and -i. It seems to me that in your contour integration you use the wrong pole. – M. Wind Nov 16 '14 at 04:21
  • Do you know the Fourier inversion theorem? Using it, you can skip the entire calculation. – PhoemueX Nov 16 '14 at 08:27

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