Complex Integral Homework: Calculate (z+(1/z))^n dz

[physics_fun]

Homework Statement

Homework Equations

I hope there's someone who can help me with the following:

I have to calculate the integral over C (the unit cicle) of (z+(1/z))^n dz, where z is a complex number.

The Attempt at a Solution

I tried to use the subtitution z=e^(i*theta), so you get
(z+(1/z))^n dz=(2*i*Sin(theta))^n * i*e^(i*theta) dtheta
but then I get stuck.
Is this the right way, and if, how do I proceed. And if it isn't, how should I do it?

 

[mjsd]

i suppose you integral looks like:
\int_{|z|=1} (z+1/z)^n dz = i \int_0^{2\pi} (e^{i\theta}+e^{-i\theta})^n e^{i\theta} d\theta

now did what you did then also try to expand the remaining e^{i\theta}=\cos \theta +i \sin \theta, and now you end up with two integrals with just cos and sin... you can then do the integral for two cases n odd and n even... etc...

 

[physics_fun]

Ok, but then you get:

(2*cos(theta))^n *e^(i*theta)

but I don't know how to get rid of the n...

(Don't know to use latex...)

 

[mjsd]

i said to use e^{i\theta}=\cos \theta +i \sin \theta to expand the second exponential.. and then multiply out to get something like
\cos^{n+1} \theta + \cos^n \theta \sin \theta and now you can try integrate these assuming that n is an integer. I am guessing that there will be two cases: n odd an n even