#algebra 

## Integration with residues example
![[Pasted image 20251128194309.png]]

## Definition
Residue of function $\LARGE f(z)$ at point $\LARGE z_0$ (which is an [[Types of singularities|isolated singularity]]) is the coefficient $\LARGE c_{-1}$ of the [[Laurent series]] for $\LARGE f(z)$ at $\LARGE z_0$:
$$\LARGE f(z)=...+ \frac{c_{-1}}{z-z_0}$$
$$\LARGE \underset{z=z_0}{\text{res}}f(z)=c_{-1}$$

## Residue theorem
for a [[Holomorphic function]] $\LARGE f(z)$ inside a closed non-self-intersecting contour $\LARGE \gamma$:
$$\LARGE \oint_\gamma f(z)dz=2\pi i\sum_{z_i \in \Gamma}\underset{z=z_i}{\text{res}}f(z)$$