#algebra 

Suppose we have domain $\LARGE D$, function $\LARGE f(z)$ is [[Holomorphic function|analytic]] in $\LARGE D$. Suppose we have a closed contour $\LARGE \gamma$ inside $\LARGE D$.
![[Pasted image 20251028164919.png]]
Then the [[Complex integration|integral]] $\LARGE \displaystyle \oint_\gamma f(z)dz=0$

Proof:
 $\LARGE f=u+iv,\space dz=dx+idy$

Then
$$\LARGE  \oint_\gamma=\oint_\gamma(udx-vdy)+i\oint_\gamma (udy+vdx)$$
Using Green's formula ($\LARGE \displaystyle\oint Pdx+ Qdy = \iint (\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})dxdy$   )

However, since the function $\LARGE f(z)$ is analytic, [[Cauchy-Riemann equations]] apply, and then:
![[Pasted image 20251028165418.png]]

Important consequence:
$\LARGE I=\displaystyle \oint_\gamma f(z)dz$
($\LARGE f(z)$ isn't necessarily analytic inside the contour)

The integral stays the same for any deformation of the contour, so long as the deformation doesn't touch the singularities of the function inside the contour
![[Pasted image 20251028165650.png]]