# Complex Analysis

I have two questions on complex integration, and I do not know how to solve them. Please help if you can.

Thanks

1. Evaluate the following principal value integral using an appropriate contour.

Integration of (integral goes from 0 to infinity) : (x)^a-1/1-x^2,
0<a<1.

2.Using contour integration and calculus of residues, find the sum

Summation (going from 0 to infinity) 1/n^2 +a^2

## Answers and Replies

shmoe
Science Advisor
Homework Helper
Hi, you'll get the most help if you provide whatever work you have, even if it doesn't appear to lead anywhere.

1) Have you seen a similar problem, but with no pole on the path of integration?

2) For series like this, a standard approach is to relate the sum to the poles of cot(Pi*z) multiplied by the appropriate function. Have you seen this method before and if so what happens when you try to apply it here?

jays said:
I have two questions on complex integration, and I do not know how to solve them. Please help if you can.

Thanks

1. Evaluate the following principal value integral using an appropriate contour.

Integration of (integral goes from 0 to infinity) : (x)^a-1/1-x^2,
0<a<1.

Show some work first.

Here are some tips :

1) Do you know the theorems that you need to use the appropriate contour ?

2) This is an analytic multivalued function (because of the a exponent) so be sure to use the branch cut. Do you know about this ?

marlon