Results 1 to 1 of 1

Math Help - contour integral around $\infty$

  1. #1
    Newbie
    Joined
    Nov 2012
    From
    italy
    Posts
    9

    contour integral around $\infty$

    Consider the function

    f(z)=Log(\frac{z-a}{z-b})

    where a,b\in D(0,r) , the disc of radius r centered at the origin, open, and r>0. Show that f is holomophic in an open subset of complex plane, containing the complement of the disc: \mathbb C-D(0,r) and compute the integral

    \oint_{|z|=r}z^nf(z)dz

    where n\geq 0 is an integer.

    I'm quite confused about this problem. Because, the $Log$ in the RHS should be the principal branch, i.e. Log(z)=log|z|+iArg(z), with Arg(z)\in (-\pi,\pi) and Arg(0)=0. Then i should prove that \frac{z-a}{z-b} does not intersect the real negative axis, but how to do this? However, suppose i could prove the holomorphicity, then i'm left to compute the integral, for which i tought to use the residue formula applied to \infty point, so calling I the integral,

    I=2\pi i Ind(\gamma,\infty)Res(f,\infty)

    where \gamma is the contour |z|=1, so that Ind(\gamma,\infty)=-1. But then the computation of residue is very hard for me. Some help?
    Last edited by tenderline; November 29th 2012 at 06:46 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: November 28th 2012, 04:27 PM
  2. contour integral, limiting contour theorem with residue
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: May 23rd 2011, 10:00 PM
  3. Replies: 2
    Last Post: March 1st 2011, 03:04 AM
  4. Replies: 2
    Last Post: August 31st 2010, 07:38 AM
  5. [SOLVED] Though contour integral
    Posted in the Calculus Forum
    Replies: 4
    Last Post: July 6th 2010, 07:54 AM

Search Tags


/mathhelpforum @mathhelpforum