# MATH3401

Lecture 33

### Examples of isolated singularities

(1) $$\,f(z)=\ds\frac{\sin z}{z}$$: $$z=0$$ is a removable singularity.

$g(z)= \left\{\array { \frac{\sin z}{z} & z\neq 0 \cr 1 & z=0 } \right.$ is entire and $$g(z) = \ds \sum_{n=0}^{\infty}\frac{(-1)^nz^{2n}}{(2n+1)!}$$. Also note $$\underset{z=0}{\res} g(z) = 0$$.

### Examples of isolated singularities

(2) $$\,f(z)=\ds\frac{1}{z^4}$$: analytic on $$\C_*$$, isolated singularity at $$z=0$$.

This is a pole of order 4, (largest negative power occuring).

Here $$\underset{z=0}{\res} f(z) = 0$$ (coefficient of the $$z^{-1}$$ term).

### Examples of isolated singularities

(3) $$\,h(z)=\ds\frac{\sinh z}{z^4}$$: analytic on $$\C_*$$, isolated singularity at $$z=0$$.

On $$\C_*$$: $h(z) = \frac{1}{z^4}\left( z+\frac{z^3}{3!}+\frac{z^5}{5!}+\cdots \right)$ $\;=\frac{1}{z^3}+\frac{1}{3!z}+\frac{z}{5!}+\cdots$

Pole of order 3 at $z=0$; here $$\underset{z=0}{\res} h(z) = \frac{1}{3!}=\frac{1}{6}$$.

### Examples of isolated singularities

(4) $$\,f(z)=e^{1/z}$$: analytic on $$\C_*$$, isolated singularity at $$z=0$$.

On $$\C_*$$: $f(z) =\ds \sum_{n=0}^{\infty}\frac{1}{n!z^n}$ Essential singularity at $z=0$. Here $$\underset{z=0}{\res} f(z) = 1$$.

### Picard's Theorem

Any analytic function with an essential singularity at $$z_0$$ takes on all possible complex values (with at most a single exception) infinitely often in any neighbourhood of $$z_0$$.

### Examples of isolated singularities

(5) $$\,f(z)=\ds\frac{1}{1-\frac{1}{z}}$$, $$z\neq 0$$: isolated singularity at $$z=0,1$$.

For $$z\neq 0,1$$: $$\quad f(z) =\ds \frac{z}{z-1}$$ $$=\ds \frac{-z}{1-z}$$ $=-z\left(1+z+z^2+\cdots \right) \qquad\quad\;\;$ $=-z-z^2-z^3-z^4-\cdots \qquad\quad\;\;$ Removable singularity at $z=0$. Taking $$f(0)=0$$ extends $$f$$ to an anlaytic function on $$\{z:|z|\lt 1\}$$. What happens at $$1$$? 📝

### Residues at Poles of order $$m$$

Theorem 1: An isolated singularity $$z_0$$ of $$f$$ is a pole of order $$m\ge 1$$ at $$z_0$$, iff $$f$$ can be written near $$z_0$$ as $f(z) = \frac{\phi (z) }{(z-z_0)^m}$ where $$\phi$$ is analytic on some $$B_{R}(z_0)$$ and $$\phi(z_0) \neq 0$$.

### Residues at Poles of order $$m$$

Theorem 2: In this case $\underset{z=z_0}{\res} f =\frac{\phi^{(m-1)}(z_0)}{(m-1)!}. \quad(*)$ In particular, for $$m=1$$ (simple pole): $\underset{z=z_0}{\res} f =\frac{\phi^{(0)}(z_0)}{0!} = \phi(z_0). \quad(**)$