# Residue at $z=0$ of $f(z)=\frac{\sin z}{1-\mathrm{e}^{1 / z}}$

I'm working on the following past paper question: In each of the following cases, $$f$$ has a singularity at 0 . Classify these singularities, and, in the case of the isolated singularities, calculate the residues:

(i) $$f(z)=\frac{\sin z}{1-\mathrm{e}^{-z}}$$;

(ii) $$f(z)=\frac{\sin z}{\left(1-\mathrm{e}^{-z}\right)^3}$$;

(iii) $$f(z)=\frac{\sin z}{1-\mathrm{e}^{1 / z}}$$.

I've completed the first two parts by computing the Laurent series and finding the $$\frac{1}{z}$$ coefficient. However, I'm struggling to see what to do for (iii). I believe $$z=0$$ is an essential singularity, but how would I compute the residue?

I've also tried it on WolframAlpha, but it doesn't seem to know what to do at $$z=0$$. Some help would be much appreciated!

Also more generally, does a function with an essential singularity have a residue/Laurent expansion?

In general, the residue of a complex function, being holomorphic at the neighborhood of a point $$z=z_0$$ is the defined as the coefficient of the term $$\frac{1}{z}$$ in its Laurent series expansion. For example: $$e^\frac{a}{z}=1+\frac{a}{z}+\frac{a^2}{2z^2}+\frac{a^3}{6z^3}+\cdots \implies \text{Rez}(e^{\frac{1}{z}})|_{z=0}=a.$$ The holomorphism condition of a complex function at the neighborhood of a point is fundamental. The problem with part C of your question is that $$\frac{\sin z}{1-e^\frac{1}{z}}$$ is not holomorphic at any neighborhood of $$z=0$$. The reason is that this function, has simple singularities at $$z=\frac{1}{2k\pi i}$$ for $$k\in \Bbb Z-\{0\}$$ and any neighborhood of $$z=0$$ will contain infinitely many of these simple singularities. So, no residue is defined for this function at $$z=0$$.
• You're welcome. 1) Yes. It does not exist. The reason is that the closed integrals defining the coefficients of the Laurent series, enclose infinitely many singularities. 2) Yes. $z=0$ is essential, but not isolated (i.e. it is not the only singularity of the function in at least one of its neighborhoods.) 3) The residue at a singularity is defined only if it is isolated. For example, $z=0$ is an essential singularity for both $text{Ln} z$ and $e^\frac{1}{z}$. However, it is an isolated singularity only for the second function, for which the residue is equal to $1$. Commented Jan 9, 2023 at 17:57
• Good luck! Commented Jan 9, 2023 at 19:47