#algebra 

## Isolated singularity
Isolated singularity of $\LARGE f(z)$ is:
$$\LARGE z_0,\space\space\space U:0<|z-z_0|<r$$
and $\LARGE f(z)$ is [[Holomorphic function|analytic]] on $\LARGE U$. (U is a [[Окрестность точки|punctured neighbourhood of z_0]])

3 types of the corresponding [[Laurent series]] in the disk exist:
### Removable singularity
$\LARGE f(z)=c_0+c_1(z-z_0)+c_2(z-z_0)^2 + ...$

(no terms with negative powers)

so that $\LARGE z_0$ - point of analyticity. 

Example:
![[Pasted image 20251128142246.png]]
(limit exists in z=0, we can additionally define f(0)=1 to make it fully analytic)

### Pole of order $\LARGE n$
$$\LARGE f(z)=\frac{c_{-n}}{(z-z_0)^n}+...+c_0+c_1(z-z_0)+...$$

(finite amount of terms with negative powers)


Example:
![[Pasted image 20251128142720.png]]

### Essential singularity
$$\LARGE f(z)=\space...+\frac{c_{-1}}{z-z_0}+c_0+c_1(z-z_0)+...$$

(infinite amount of terms with negative powers)

![[Pasted image 20251128142914.png]]

(essential singularity whenever we have an exponential function in the infinite power)
e.g.:
![[Pasted image 20251128143231.png]]

### Difference between poles and essential singularities
![[Pasted image 20251128142944.png]]
![[Pasted image 20251128142956.png]]
![[Pasted image 20251128143023.png]]

## Non-isolated singularity
![[Pasted image 20251128143329.png]]


## Branch point
![[Pasted image 20251128143354.png]]