Solving Complex Integration: Principal Value and Summation with Contour Methods

[jays]

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

 

[shmoe]

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?

 

[marlon]

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