Help calculating Residue at an essential singularity

Question

Can u help me calculate the residue of the function $f(z)= (z^2/(1+9z^2))(e^{i/z}-1)$ at 0? Ive tried using the Laurent series but its seems very confusing to figure it out, and ive tried to use the residue Theorem around a circle of radius epsilon but i cant seem to figure it out,thx in advance.

Answer 1

You could multiply the power series for both factors.

Or use that the sum over all residua at $0,\pm\frac i3$, $\infty$ is zero.

Or that the residua sum over the finite singularities is $$ \lim_{R\to\infty}\int_{|w|=1}\frac{R^2w^2}{1+9R^2w^2}(e^{i/(Rw)}-1)\,d(Rw) = \lim_{\varepsilon\to0}\int_{|w|=1}\frac{w^2}{ε^2+9w^2}\frac{e^{iε/w}-1}ε\,dw. $$