Fourier Transform of $S_\epsilon(x)=e^{−2π\epsilon|x|}\operatorname{sgn} x$

Question

I need to calculate the Fourier transform of $S_\epsilon(x)=e^{-2\pi\epsilon|x|}\operatorname{sgn} x$. I tried by definition and got stuck. Can anyone help me or give a hint to solve the problem?

Answer 1

This is mechanised in CASes. For example, the command of Mathematica FourierTransform[ Exp[-2*Pi*\[Epsilon]*RealAbs[x]]*Sign[x], x, \[Omega]] answers $$\frac{i \sqrt{\frac{2}{\pi }} \omega }{\omega ^2+4 \pi ^2 \epsilon ^2} .$$

Answer 2

I think it's worthwhile to know the relatively simple way to directly evaluate this by hand. Namely, let $f(x)$ be $0$ for $x<0$ and $e^{-hx}$ for $x>0$. The Fourier transform is $$ \int_{-\infty}^\infty e^{-2\pi i\xi x}\;f(x)\;dx \;=\; \int_0^\infty e^{-2\pi i\xi x} \;e^{-hx}\;dx \;=\; \int_0^\infty e^{(-2\pi i\xi -h)x}\;\;dx $$ $$ \;=\; {-1\over 2\pi i\xi + h} \Big[e^{(-2\pi i\xi -h)x}\;\;\Big]_0^\infty \;=\; {1\over 2\pi i\xi + h} $$ A similar direct computation applies to the left-hand half of your function.