Fourier Transformation of $e^{-a|x|}$

Question

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?

Answer 1

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 $x\gt 0$: $$ F_\text{trans}= \int_{-\infty} ^{\infty} f(x) e^{-ikx}\,dx = \int_{-\infty} ^{\infty} e^{-a|x|} e^{-ikx}\,dx \\ = \int_{-\infty} ^{0} e^{ax} e^{-ikx}\,dx + \int_{0} ^{\infty} e^{-ax} e^{-ikx}\,dx \\ = \int_{-\infty} ^{0} e^{(a-ik)x}\,dx + \int_{0} ^{\infty} e^{-(a+ik)x}\,dx. $$