# Evaluate integral using Cauchy's Thm

• May 5th 2010, 11:09 PM
jzellt
Evaluate integral using Cauchy's Thm
Evaluate:

z
------- dz
(z-2)^2

and we're integrating from |z|=1. I'm asked to evaluate this using Cauchy's Thm, but I don't know how

Can someone show the steps please. Thanks a lot
• May 5th 2010, 11:31 PM
chisigma
The Cauchy's theorem extablishes that...

$\int_{\gamma} f(z)\cdot dz = 2\pi i \cdot \sum_{n} R_{n}$ (1)

... where $R_{n}$ is the residue of each pole of f(*) inside $\gamma$. In Your case $f(z)= \frac{z}{(z-2)^{2}}$ and $\gamma$ is the unit circle... how many poles of f(*) are inside $\gamma$?...

Kind regards

$\chi$ $\sigma$
• May 9th 2010, 08:52 PM
jzellt
I know what Cauchy's theorem says, but I don't know how to evaluate an integral using it.

Can someone please post the steps showing how to do this?

(I not here asking you guys to do my hw for me. I'm going to have a similiar question on my final and would like know how to go about this type of problem)

• May 10th 2010, 03:27 AM
mr fantastic
Quote:

Originally Posted by jzellt
I know what Cauchy's theorem says, but I don't know how to evaluate an integral using it.

Can someone please post the steps showing how to do this?

(I not here asking you guys to do my hw for me. I'm going to have a similiar question on my final and would like know how to go about this type of problem)

Do you know what a pole is? Do you know what a residue is? Do you know how to calculate a residue? These are things you need to know before attempting to use the Residue Theorem to integrate.

Also, Cauchy's Integral Formula says:

If f(z) is analytic in a region R containing a closed contour C and $\alpha$ is any point enclosed by C then $f^{(n)}(\alpha) = \frac{n!}{2 \pi i} \oint_C \frac{f(z)}{(z - \alpha)^{n+1}} \, dz$.

However, all this is irrelevant to your present question since the function you're integrating is analytic inside and on the closed contour |z| = 1 and so the value of the integral is zero.
• May 10th 2010, 03:30 PM
jzellt
Thanks for the input. Yes, I know what a pole and residue are...

But I'm still not sure how to evalute this integral using Cauchy...

Can someone show how? Thanks