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!

Application of Cauchy's residue theorem

  1. Aug 21, 2013 #1
    Really need help for this one. Cheers!




    1. The problem statement, all variables and given/known data

    Question: calculate function z/(1-cos z) integrated in ac ounterclockwise circular contour given by |z-2pi|= 1

    2. Relevant equations



    3. The attempt at a solution

    Clearly the pole in the given contour is 2pi. But the problem is: if it's a simple pole, then apply formula

    we have Residue=lim (z-2pi) * z/(1-cos z) where z->2pi. This limit does not exist.

    So I reckon 2pi might be a higher order pole but this actually makes no sense and even if it's true,

    there is a really nasty differentiation.

    Any thoughts.?
     
    Last edited: Aug 21, 2013
  2. jcsd
  3. Aug 21, 2013 #2
    Please use the standard format of PF!
     
  4. Aug 21, 2013 #3

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    Yes, please use LaTeX.

    What you need is a Laurent expansion around the pole at [itex]z=2 \pi[/itex]. For your function
    [tex]f(z)=\frac{z}{1-\cos z}[/tex]
    you have
    [tex]1-\cos z=\frac{(z-2 \pi)^2}{2}+\mathcal{O}[(z-2 \pi)^4].[/tex]
    You need the single-order pole contribution. You can get this by either using the standard formula
    [tex]\text{res}_{z \rightarrow 2 \pi} f(z)=\lim)_{z \rightarrow 2 \pi} \frac{\mathrm{d}}{\mathrm{d} z} (z-2 \pi)^2 f(z),[/tex]
    which is a bit cumbersome here, or you use the series expansion of [itex]1-\cos z[/itex] as given above and use [itex]z=(z-2 \pi)+2 \pi[/itex] in the numerator.
     
  5. Aug 21, 2013 #4
    sorry for not using LaTex, havent used this forum in ages.

    And thanks so much for the help! very smart move!!

    Correct me if im wrong, the answer is 4i*pi?
     
    Last edited: Aug 21, 2013
  6. Aug 22, 2013 #5

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    Looks good!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Application of Cauchy's residue theorem
  1. Cauchy Residue Theorem (Replies: 12)

Loading...