Using generalized Cauchy Theorem(complex analysis)


MHF Hall of Honor
Aug 2008
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:

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