How to compute the residue of a complex function with essential singularity I'm a student of mechanical engineering and I have a problem with computing residues of a complex function. I've read some useful comments in the other threads. Now I've got some ideas about essential singularity and series expansion in computing the residue. However, I still can't find the solution to my problem.
I arrived at a complex function in the process of finding a solution to a mechanical problem.
Then I had to obtain the residues to proceed to the next steps. The function has the following form:
$$f(z)=\frac{\exp(Az^N+Bz^{-N})}{z}$$
where $A$, $B$ and $N$ are real constants $(N \geq 3)$.
I want to compute the resiude at $z=0$. I wrote the Laurent series of $f$, but got an infinite sum. I do not even know if I am at the right direction.
I would be really thankful if someone could give me a hint on this and put me back in the right direction.
 A: From each even term in the exponential series, you get one contribution where the positive and negative powers of $z$ cancel out. This is the middle term of the binomial expansion, and to get the coefficient $a_{-1}$ in the Laurent series you need to sum this over all even terms, which leads to
$$a_{-1}=\sum_{k=0}^\infty\frac1{(2k)!}\binom{2k}{k}A^kB^k=\sum_{k=0}^\infty\frac{(AB)^k}{k!^2}\;.$$
A: I'm assuming $N$ is an integer here to avoid issues with branches of the logarithm.
Note that $\exp(Az^N) = \sum_{m=0}^{\infty} {A^mz^{mN} \over m!}$ and $\exp(Bz^{-N}) = \sum_{m=0}^{\infty} {B^mz^{-mN} \over m!}$, so that $\exp(Az^N + Bz^{-N})$ is the product of these two series, and ${\displaystyle{\exp(Az^N + Bz^{-N}) \over z}}$ is the product of these two series divided by $z$.
The residue at $z = 0$ will be the cofficient of the ${1 \over z}$ term in this product divided by $z$, or in other words the constant term in the product. A constant term is obtained in the multiplication when you multiply a term ${A^mz^{mN} \over m!}$ by a term ${B^mz^{-mN} \over m!}$ (with the same $m$). So the constant term in the overall product is the sum of these, that is, $\sum_{m=0}^{\infty} {(AB)^m \over (m!)^2}$ and this will be your residue.
