# Thread: Using generalized Cauchy Theorem(complex analysis)

1. ## 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

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