What is the best approach to evaluate the contour integral:
$$\int_{-\infty}^{\infty} \frac{e^{ibx}}{x^2+a^2} dx$$
in the case that a is complex, $Re(a) > 0$ and b is positive?
If $a$ was real, I can determine the result by using a semicircular contour,
applying Jordan's Lemma along the arc, and then using the residue theorem to
compute the residue at z= ia. I assume that I need to a similar thing here, but
I'm having trouble actually determining the residues and when they are inside
the contour. If I write $a = a_{1}+ia_{2}$ then I have:
$a^{2} = a_{1}^{2}+2ia_{1}a_{2}-a_{2}^{2}$ so I'm looking for the roots of:
$z^{2}+a_{1}^{2}+2ia_{1}a_{2}-a_{2}^{2}= 0$. Would I proceed by writing
z and a in polar form? I feel like there is an easier way to do this. Any
suggestions would be greatly appreciated.
Best Answer
The roots of $z^2+a^2$ are $\pm ai$ and$$\operatorname{res}_{z=\pm ai}\frac{e^{ibz}}{z^2+a^2}=\frac{e^{ib\times(\pm ai)}}{\pm2ai}=\pm\frac{e^{\mp ab}}{2ai}=\mp\frac{e^{\mp ab}}{2a}i.$$