Fourier transform of $f(x)= e^{-a|x-b|}$ where b is a parameter? I needed to compute the Fourier transform of the function in the title. Fourier transform of $e^{-a|x|}$ is ok for me so I wanted to use a new variable $u=x-b$ but that apparently won’t do it and apparently there should be a domain where the transform is forbidden. 
Thanks for your help!
 A: The Fourier transform of a "symmetric exponential" function is a Cauchy-Lorentz function :
$$\dfrac{1}{\sqrt{2 \pi}}\int_{-\infty}^{+\infty}e^{-i\omega x}e^{-a|x|}dx=\dfrac{\sqrt{2}}{\sqrt{\pi}}\dfrac{a}{(a^2+\omega^2)}\tag{1}$$
Replacing $x$ by $x-b$ in the LHS of (1) gives 
$$\dfrac{1}{\sqrt{2 \pi}}\int_{-\infty}^{+\infty}e^{-i\omega (x-b)}e^{-a|x-b|}dx=\dfrac{\sqrt{2}}{\sqrt{\pi}(a^2+\omega^2)}\tag{2}.$$
Extracting $e^{i \omega b}$ form the integral on the LHS gives, and multiplying both sides by $e^{-i \omega b}$ finally gives :
$$\dfrac{1}{\sqrt{2 \pi}}\int_{-\infty}^{+\infty}e^{-i\omega x}e^{-a|x-b|}dx=\dfrac{\sqrt{2}e^{-i \omega b}}{\sqrt{\pi}(a^2+\omega^2)}\tag{3}.$$
meaning that the F.T. of $x \to e^{-a|x-b|}$ is $\dfrac{\sqrt{2}e^{-i \omega b}}{\sqrt{\pi}(a^2+\omega^2)}.$
Remark : we retrieve here one the properties of the F.T. (Fourier Transform) that transforms a translation by $b$ in the space (or time) domain by a multiplication by $e^{i \omega b}$ in the frequency domain.
If the definition of FT I have used in (1) is not yours, convert pulsation $\omega$ into $2 \pi \nu$ where $\nu$ is frequency (and multiply the result by $\sqrt{2 \pi}$).
