#algebra 

## Definition
Suppose we have some domain, and a contour $\LARGE \gamma$ inside.
Inside the contour we have some point $\LARGE a$, then:

$$\LARGE f(a)=\frac{1}{2\pi i}\oint_\gamma \frac{f(z)}{z-a}dz$$
![[Pasted image 20251028201206.png]]
Essentially, a [[Holomorphic function]] defined on a disk is completely determined by its values on the boundary of the disk

## Proof

We can't use [[Cauchy's integral theorem]] straight away because in point $\LARGE z=a$ we have a singularity, so we have to modify our contour:
![[Pasted image 20251028201237.png]]
infitesimal circle of radius $\LARGE \varepsilon$, and two infintely close linear segments $\LARGE l_1$ and $\LARGE l_2$.
New contour: $\LARGE \gamma_1 = \gamma+ l_2+\varepsilon+l_1$ 
![[Pasted image 20251028201337.png]]

Then:
$$\LARGE \oint_{\gamma_1}\frac{f(z)}{z-a}dz=0$$
$$\LARGE \oint_{\gamma_1}=\oint_\gamma+\int_{l_2}+\int_{l_1}+\int_\varepsilon$$
Integrals on $\LARGE l_1$ and $\LARGE l_2$ cancel each other out

The remaining integral can be easily computed:
![[Pasted image 20251028201656.png]]

### Generalization for an annulus
![[Pasted image 20251028201900.png]]

Just add 4 linear segments:
![[Pasted image 20251028201923.png]]
![[Pasted image 20251028202006.png]]