What is the purpose of this manipulation?

Question

In my complex analysis class, we often perform a transformation to investigate the singularities of a function. For example, given $f(z)=\cot(1/z)=\frac{\cos(1/z)}{\sin(1/z)}$ where we find that $f(z)$ has the singularties $z=0$ and $z=\frac{1}{n\pi}$. We then let $\frac{1}{z}=u+n\pi$, plug it into the function, and perform various manipulations to get the expression into a series form where we can classify the singularities based on the characteristics of the series. My question is why are we letting $\frac{1}{z}=u+n\pi$, and what does it do to aid in such an analysis?

Answer 1

The singularities of your function are created when $\sin(\frac 1z)=0$.

$\sin(\theta)$ is a periodic function with zeroes at $\theta = 0, \pi, 2\pi, 3\pi, ...$

It is easier to describe these as being at $\theta = n\pi$.

What I have been describing as $\theta$ are in your case the $\frac 1z$, so your singularities are at $\frac 1z=n\pi$.

You want to study the behaviour close to the singularities, so you consider $\frac 1z=n\pi+$"a little bit", or (more mathematically) $\frac 1z=n\pi+u$. After finding the series for $f(z)$ in terms of $u$ you can then consider what happens to this series as $u$ approaches zero, ie as you get closer to the singularity.

You asked: So how would I investigate the specific singularity $z=0$? Would I allow $u$ to approach $−n\pi$?

No, this is a completely different kind of singularity. Try $z=u$ and see how that turns out...