Under which conditions is $\int_\Gamma f=0$

Question

Let $\Gamma$ be a contour. $V\subseteq \mathbb C$ and $f:V\rightarrow \mathbb C$.

What are the conditions on:

• $\Gamma$
• $f$
• Open $U\subseteq V$

such that $\int_\Gamma f=0$

This seems like an open mapping theorem which states that open sets get mapped to open sets.

I thought:

Let $f$ be holomorphic everywhere, let $\Gamma$ be a closed loop. $U$ to be anything. Is this correct?

Answer 1

Look for the Cauchy integral theorem and you will get your answer

Answer 2

Integral around contours are computed with the residue formula. So the sum of the residues inside the contour must be zero. Take the Laurent development at each singular point in the contour and compute it's residue.