TITLE: Fourier Transformation of $e^{-a|x|}$
QUESTION [6 upvotes]: The Fourier Transformation will be like this : 
$$ F_\text{trans}= \int_{-\infty} ^{\infty} f(x) e^{-ikx} dx  =  \int_{-\infty} ^{\infty} e^{-a|x|} e^{-ikx} dx $$
I don't understand what to do after that line. How will I deal with the the modulus  x?

REPLY [12 votes]: Split the integral into two regions and use the fact that $\vert x\vert = -x$ for $x \lt 0$ and $\vert x\vert = x$ for $x\gt 0$:
$$
F_\text{trans}= \int_{-\infty} ^{\infty} f(x) e^{-ikx}\,dx  =  \int_{-\infty} ^{\infty} e^{-a|x|} e^{-ikx}\,dx \\
= \int_{-\infty} ^{0} e^{ax} e^{-ikx}\,dx + \int_{0} ^{\infty} e^{-ax} e^{-ikx}\,dx \\
= \int_{-\infty} ^{0} e^{(a-ik)x}\,dx + \int_{0} ^{\infty} e^{-(a+ik)x}\,dx.
$$