1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Finding residue of a function(complex analysis)

  1. May 12, 2012 #1
    1. The problem statement, all variables and given/known data
    I need to find the residue of [itex]\frac{e^{iz}}{(1+9z^{2})^{2}}[/itex] so I can use it as part of the residue theorem for a problem.

    2. Relevant equations
    Laurent Series
    R(z[itex]_{0}[/itex]) = [itex]\frac{g(z_{0})}{h^{'}(z_{0})}[/itex]

    3. The attempt at a solution
    I tried using the laurent series but after expanding I got a 0 in the denominator.
    For the second equation I used I also got a 0 in the denominator and I don't believe it converges. Any help would be appreciated. Thanks!
  2. jcsd
  3. May 12, 2012 #2
    Look at the inverse of the function (call it f(z)), you have a zero of order 2. So you have a pole of order 2 for your function. The formula for this goes like

    ## \text{Res}(f,i/3) = \lim_{z \to a}\frac{d}{dz}\left( (z-i/3)^2 \frac{\exp (i z)}{(1+9z^2)^2} \right) ##

    you can do the others

    ## \text{Res}(f,a) = \lim_{z \to a} \frac{1}{(m-1)!}\frac{d^{m-1}}{dz^{m-1}} (z-a)^m f(z) ##

    If the pole is of order m. I think. Maybe look that up.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook