Can anyone solve these two integrals .
$$ \int_{0}^{ \infty } \frac{x^2 e^{-x^2/2 \sigma ^2}}{(x-a)^2+b^2} dx $$
and
$$ \int_{0}^{ \infty } \frac{e^{-(\ln x - \mu )^2/2 \sigma ^2}}{(x-a)^2+b^2} dx $$
Either give me some hint or tell me the procedure, if anybody knows how to solve these type of exponential integrals.
Thanks
I'm going to assume that $a$ and $b$ are both real numbers and that $b \neq 0$. Then the function $\frac{z^{2}e^{-z^{2}/\sigma^{2}}}{(z-a)^{2} + b^{2}}$ for complex-valued $z$ is meromorphic with poles at $z = a \pm bi$.
Then my idea would be to try integrating over a quarter pie-slice contour: $\Gamma_{R} = [0,R] \cup i[0,R] \cup \{ Re^{i\theta} \, : \, \theta \in [0,\pi/2]\}$, oriented counter-clockwise. Then apply the Cauchy residue formula, and if one can show the integrals over the second and third components of $\Gamma_{R}$ go to zero as $R \to \infty$, we are done.