Question 5

Question

Cauchy’s theorem and integral formula.

Answer

Cauchy’s theorem

Let $f:D\to \mathbb{C}$ be holomorphic.

Suppose ${\gamma }_1$ and ${\gamma }_2$ are two (parametrized) closed curves in $D$ such that ${\gamma }_1\sim {\gamma }_2$. Then we have
${\int }_{{\gamma }_1}f={\int }_{{\gamma }_2}f$

Cauchy’s Integral Formula

Let $C=\{z\in \mathbb{C}:|z-z_0|=r\}$ be the circle radius $r$ centred at $z_0\in \mathbb{C}$, with positive orientation. Then
${\oint }_C\frac{1}{z-z_0}dz=2\pi i$