# Prove that $\int_{\gamma} f(z)\ dz = 0$ for any closed contour $\gamma$ in the region $G$.

Prove that if $f$ is analytic in a region $G$ then $\int_{\gamma} f(z)\ dz = 0$ for any closed contour $\gamma$ on the region $G$.

I have proved this result for open disk but fail to generalize it for any region. Please help me.

Thank you in advance.

• This is only valid for simply connected regions. Sep 2, 2017 at 10:35
• Would you please explain why? I have much difficulty in understanding it. Please help me.
– user251057
Sep 2, 2017 at 10:38
• You are aware that $\int_C dz/z=2\pi i$ when $C$ is the unit circle? Sep 2, 2017 at 10:39
• Yes I know it. Actually I want to know that what is the special characteristics of simply connected domain for having some smooth properties.
– user251057
Sep 2, 2017 at 10:43
• If you have Stein's Complex Analysis at hand, check Theorem 5.1 in Chapter 3. The main point is: in a simply connected region all curves that start from $z_0$ and ends at $z_1$ can be transformed continuously into each other, thus equating the integral of any holomorphic function along them.
– Vim
Sep 2, 2017 at 10:51