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.

  • $\begingroup$ This is only valid for simply connected regions. $\endgroup$ Sep 2, 2017 at 10:35
  • $\begingroup$ Would you please explain why? I have much difficulty in understanding it. Please help me. $\endgroup$
    – user251057
    Sep 2, 2017 at 10:38
  • 1
    $\begingroup$ You are aware that $\int_C dz/z=2\pi i$ when $C$ is the unit circle? $\endgroup$ Sep 2, 2017 at 10:39
  • $\begingroup$ Yes I know it. Actually I want to know that what is the special characteristics of simply connected domain for having some smooth properties. $\endgroup$
    – user251057
    Sep 2, 2017 at 10:43
  • $\begingroup$ 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. $\endgroup$
    – Vim
    Sep 2, 2017 at 10:51


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