I have this function $$ f(z) = \frac{e^{1/z}}{ 1 – z^2} $$ which has two poles of order $1$ with residues $- e / 2$ and $ 1 / (2e)$. Now I need to compute the residue at the essential singularity $z = 0$ where the numerator becomes undefined. I expanded this in series: \begin{align*} f(z) = \frac{1}{- (z + 1) (z – 1)} \big( 1 + \frac{1}{z} + \frac{1}{2} \frac{1}{z^2} + \ldots \big). \end{align*} But I'm not sure how to find the residue from this.
Complex Analysis – Residue of an Essential Singularity
complex-analysis
Best Answer
You can easily compute the sum of all three residues as
$$\frac{1}{2\pi i} \int_{\lvert z\rvert = R} \frac{e^{1/z}}{1-z^2}\,dz$$
with $R > 1$. Then just subtract the residues at the poles from the sum to find the residue at $0$.