Let . Find

in terms of , where is the circle oriented in the counterclockwise

direction.

I tried to express as . Then how should I proceed on? Any idea/suggestion is welcomed!:D

Printable View

- Nov 16th 2012, 06:59 PMalphabeta89Complex analysis - integrating over a contour
Let . Find

in terms of , where is the circle oriented in the counterclockwise

direction.

I tried to express as . Then how should I proceed on? Any idea/suggestion is welcomed!:D - Nov 16th 2012, 09:37 PMchiroRe: Complex analysis - integrating over a contour
Hey alphabeta.

Can you use the Cauchy-Formula and evaluate f(z) at alpha? (Since f(z) has no singularities in the interior of the contour)? - Nov 16th 2012, 10:12 PMProve ItRe: Complex analysis - integrating over a contour
First, we need to check that there are not any singularities. The only place where there may be a singularity is where . But this is not possible since on the contour , for that denominator to be 0, , but we know . Now since is the unit circle oriented in the counterclockwise direction centred at the origin, we can parameterise it as with . Notice that and the integral becomes

Now let and note that and giving

- Nov 16th 2012, 10:57 PMalphabeta89Re: Complex analysis - integrating over a contour
- Nov 16th 2012, 11:02 PMalphabeta89Re: Complex analysis - integrating over a contour
- Nov 16th 2012, 11:05 PMProve ItRe: Complex analysis - integrating over a contour
- Nov 16th 2012, 11:09 PMProve ItRe: Complex analysis - integrating over a contour
- Nov 16th 2012, 11:11 PMalphabeta89Re: Complex analysis - integrating over a contour
- Nov 16th 2012, 11:17 PMProve ItRe: Complex analysis - integrating over a contour
- Nov 16th 2012, 11:22 PMProve ItRe: Complex analysis - integrating over a contour
To answer your other question, there is a singularity at inside the region, so you will need to use the residue theorem.

, where

That limit should be very easy to evaluate...