A (probably complex) integral

This is an exercise from Stein & Shakarchi - Complex Analysis

II.7) Prove that for

How do I start?

I know this function has a pole in

I know the function can be rewritten as: or as

What I do not know: what to do? Do I use the residue theorem? Do I look for a nice contour?

(Also in general: how to tackle complex integrals? Step by step.)

Re: A (probably complex) integral

Quote:

Originally Posted by

**CSM** This is an exercise from Stein & Shakarchi - Complex Analysis

II.7) Prove that

for

How do I start?

I know this function has a pole in

I know the function can be rewritten as:

or as

That last suggestion is the best one. Write the integral as Then make the substitution so that the integral becomes an integral round the unit circle and you can use the residue theorem.

Re: A (probably complex) integral

If the function is or and the limits are or , the contour will be a circle.

Re: A (probably complex) integral

With the unit circle.

So it has a pole in ?

Re: A (probably complex) integral

I rewrote it as which only has a pole in the unit circle and for in .

This is a second order pole (?)

So I have to calculate ?

I'm asking because that will be quite ugly I guess...