• Support PF! Buy your school textbooks, materials and every day products Here!

Complex integral

  • #1

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???
 
Last edited:

Answers and Replies

  • #2
mjsd
Homework Helper
726
3
i suppose you integral looks like:
[tex]\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[/tex]

now did what you did then also try to expand the remaining [tex]e^{i\theta}=\cos \theta +i \sin \theta[/tex], 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...
 
  • #3
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...)
 
  • #4
mjsd
Homework Helper
726
3
i said to use [tex]e^{i\theta}=\cos \theta +i \sin \theta[/tex] to expand the second exponential.. and then multiply out to get something like
[tex]\cos^{n+1} \theta + \cos^n \theta \sin \theta[/tex] 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
 

Related Threads on Complex integral

  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
10
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
3
Views
997
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
7
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
1
Views
684
  • Last Post
Replies
4
Views
1K
Top