I have to evaluate the integrals $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}}$, for $p > 0$, and $\displaystyle \int_{-\infty}^{\infty} \frac{dx}{(x^{2}+p^{2})^{2}}$, for $p > 0$ using residues.
For the first integral, we have two simple poles, one at $x = -ip$ and one at $x=ip$. So, I used the formula $\displaystyle res_{a}f = res_{a}\frac{P(x)}{Q(x)} = \frac{P(a)}{Q^{\prime}(a)}$ to calculate them. (First off, is this formula correct?)
I got that $\displaystyle res_{-ip} = \frac{1}{(x^{2}+p^{2})^{\prime}}\Large \vert_{x=-ip}$ $ \displaystyle = \frac{i}{2p}$. However, according to Maple, the resdiue at $x = -ip$ is $0$. Why the discrepancy?
I wound up getting $\displaystyle res_{ip} = \frac{-i}{2p}$, so that my integral $\displaystyle \int_{-\infty}^{\infty}\frac{dx}{x^{2}+p^{2}} = 0$, because when you add the two residues that I got together, and then multiply them by $2\pi i$, you get $0$. According to Maple, that should be the value of this integral, but it would appear that I am getting there using incorrect residues. Could somebody please tell me what I'm doing wrong and how to fix it?
For the second integral, I'm getting the residues $\displaystyle -\frac{1}{4ip^{3}}$ and $\displaystyle \frac{1}{4ip^{3}}$, respectively, for the residues at the $2$nd order poles $x = -ip$ and $x=ip$, when according to Maple, I should be getting $0$ for each.
So, I am really extremely confused about what is going wrong, and I really need to know how to properly apply residues to evaluating integrals, so I need someone to correct what I am doing and steer me in the right direction.
Thank you.