Using generalized Cauchy Theorem(complex analysis)

**Hyungmin** · January 17th 2010, 11:00 AM

Q5) i know what is the generalized Cauchy Theorem but just dont know how to apply to the questions as finding the domains.

I attached the problem sheet or website

http://www.maths.manchester.ac.uk/~s...mplexprob4.pdf

So please help me

thank you in advance !

**shawsend** · January 19th 2010, 08:58 AM

Let $f(z)=\frac{1}{(z-1)(z+1)}$ and consider the three contours $\gamma=2e^{it}$ , $\gamma_1=-1+1/2e^{it}$ and $\gamma_2=1+1/2e^{it}$ for $0\leq t\leq 2\pi$ . Then $f(z)$ is analytic on all of these contours and throughout the multiply-connected domain consisting of all the points inside the larger circle $C(2,0)=\gamma$ and outside the two smaller circles $C(1/2,-1)=\gamma_1$ and $C(1/2,1)=\gamma_2$ . Therefore by the extended Cauchy-Goursat theorem:

$\int_{\gamma} f(z)dz=\int_{\gamma_1} f(z)dz+\int_{\gamma_2} f(z)dz$

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