Singular points are $0, a,\infty$ and all of them are isolated. If $n=1$, then both $0,a$ are simple poles, so calculating residues are easy and by the Residue Theorem residue at $\infty$ is also can be found. My problem is to calculate the residue at $0$ when $n>1$. What I have tried so far is the following:
$$\small\frac{1+z^{2n}}{z^n(z-a)}=\left(\frac{1}{z^n}+z^n\right)\left(-\frac{1}{a}\right)\frac{1}{1-\frac{z}{a}}=\left(\frac{-1}{z^n}-z^n\right)\sum_{n=0}^{\infty} \frac{z^n}{a^{n+1}}=\sum_{n=0}^{\infty} \frac{-1}{a^{n+1}}-\sum_{n=0}^{\infty} \frac{z^{2n}}{a^{n+1}}$$
But this expansion around $0$ implies that residue is zero. Is this correct? Am I missing something?
Also, I want to know how do we find the Laurent Expansion of this function around $\infty$?
Thanks for any help.
Best Answer
In your computations, you used the letter $n$ for two distinct things.
The residue at $0$ of $f$ is the residue at $0$ of $\frac1{z^n(z-a)}$, since $\frac{z^{2n}}{z^n(z-a)}=\frac{z^n}{z-a}$, which has a removable singularity at $0$.
Besides,\begin{align}\frac1{z^n(z-a)}&=-\frac1a\times\frac1{z^n}\times\frac1{1-\frac za}\\&=-\frac1a\times\frac1{z^n}\times\left(1+\frac za+\frac{z^2}{a^2}+\frac{z^3}{a^3}+\cdots\right)\\&=-\frac1{az^n}-\frac1{a^2z^{n-1}}-\frac1{a^3z^{n-2}}-\cdots\end{align}and the coefficient of $\frac1z$ in this Laurent series is $-\frac1{a^n}$ . In other words,$$\operatorname{res}_{z=0}f(z)=-\frac1{a^n}.$$