# Using generalized Cauchy Theorem(complex analysis)

#### Hyungmin

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/~sharp/complexprob4.pdf

So please help me #### Attachments

• 45.1 KB Views: 30

#### shawsend

MHF Hall of Honor
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:

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