# Basic Complex Analysis question

• Nov 29th 2007, 10:12 AM
thecow
Basic Complex Analysis question
This question appears really easy, I am just not sure I am undestanding it correctly.

I apologize for the large size, ha. Anyways question number 2 is what I am having difficulty with. I realize it is a closed path with poles, but the shape of the path is what is throwing me off. It seems that for the 0 for instance, you would multiply the answer basic Cauchy-Integral answer by -3, but maybe I am not understanding something?
• Nov 29th 2007, 12:42 PM
CaptainBlack
Quote:

Originally Posted by thecow
This question appears really easy, I am just not sure I am undestanding it correctly.

I apologize for the large size, ha. Anyways question number 2 is what I am having difficulty with. I realize it is a closed path with poles, but the shape of the path is what is throwing me off. It seems that for the 0 for instance, you would multiply the answer basic Cauchy-Integral answer by -3, but maybe I am not understanding something?

You need the version of the residue theorem with winding number.

RonL
• Nov 29th 2007, 01:19 PM
thecow
We havent yet gone over the residue theorem. We are just supposed to note that there are closed loops around a pole, and then figure out what the result would be. The problem is this graph is very complicated, so I am not entirely sure. The way I see it is that for instance the pole at 0, the integral would just be the integer 3. For the pole at 1, you would get 2*e^(i*pi)
• Nov 29th 2007, 02:42 PM
ThePerfectHacker
Quote:

Originally Posted by CaptainBlank
You need the version of the residue theorem with winding number.

RonL

Quote:

Originally Posted by thecow
We havent yet gone over the residue theorem. We are just supposed to note that there are closed loops around a pole, and then figure out what the result would be. The problem is this graph is very complicated, so I am not entirely sure. The way I see it is that for instance the pole at 0, the integral would just be the integer 3. For the pole at 1, you would get 2*e^(i*pi)

You can do what CaptainBlank said and use the residue theorem. But here is a special case, you can rely on the Cauchy integral formula but you still need winding numbers because $\Gamma$ is not a contour (i.e. a simple closed curve).
• Nov 29th 2007, 02:47 PM
thecow
Yea I mean, I get that its a winding number. I am just curious if I am reading it correctly, so for instance the pole at 0 would be a winding number of 3?
• Nov 29th 2007, 02:52 PM
ThePerfectHacker
Quote:

Originally Posted by thecow
Yea I mean, I get that its a winding number. I am just curious if I am reading it correctly, so for instance the pole at 0 would be a winding number of 3?

Yes. Just count the number of times it "winds" around the number.
• Nov 29th 2007, 02:52 PM
thecow
Alright thanks. I thought it might be 2 because if you follow the path with your finger you really only go around twice it seems.