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Math Help - Using generalized Cauchy Theorem(complex analysis)

  1. #1
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    Using generalized Cauchy Theorem(complex analysis)

    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 !
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  2. #2
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    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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