I am looking for solution of $\int_{-\infty}^{\infty}1/(x^{2n}-a_0)^2 dx$ for some unknown constant $a_0$. I know that some related forms have been earlier solved with the help of residue theorem. However, I find this one particularly troublesome due to the constant $a_0$.
So far, I've found that the complex poles are $z_k=|a_0|^{1/2n}\mathrm{exp}\big(j(\theta+2\pi k)/2n\big)$. And the contour integral can be written as \begin{align} \oint f(z) dz &=2\pi i \sum_{k=1}^{2n}\frac{d}{dz}(z-z_k)^2f(z) \\ &=2\pi i \sum_{k=1}^{2n}\frac{d}{dz} \frac{(z-z_k)^2}{\prod_{k=1}^{2n}(z-z_k)^2} \end{align} I am unable to go beyond this. I am just an engineer, that has found himself in midst of complex mathematical problems. I would really appreciate all the help I can get.
P.S. As an engineer with no background in complex function theory and residue theorem, I am really intrigued and drawn to the topic. Is there any good concise book on this topic that someone can recommend? Thanks
