Results 1 to 2 of 2

Math Help - Complex Integration - Resiidue Theorm

  1. #1
    Super Member
    Joined
    Apr 2009
    Posts
    678
    Thanks
    1

    Complex Integration - Resiidue Theorm

    f(z) has isolated singularity at z=z_0
    Now, \oint _Cf(z).dz = 2\pi i Res_{z0}f(z)
    where C is a closed path around z_0
    (This is directly from Residue Theorem)

    My question is what is this integral if C is rather an open semi-circular arc around z_0

    Do we have any formula/theorem for this?
    May in some special case - (I'm more interested in finding the limit of this integral when the semi-circular arc in getting closer and closer to z_0

    Can I say \oint _Cf(z).dz = \pi i Res_{z0}f(z)?

    Is this result ever applicable?

    Sorry, if my questions are vague but don't have a clear idea to ask a specific question at this moment
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Let f(z) have a simple pole at z_0 and \gamma be an arc of a circle of radius r, and angle \alpha then:

    \lim_{r\to 0} \int_{\gamma} f(z)dz=\alpha i \mathop\text{Res}_{z=z_0} f(z)

    Also, something can be said about poles of other orders. See:

    Art of Problem Solving Forum
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Complex Numbers De Moirves Theorm
    Posted in the Advanced Math Topics Forum
    Replies: 6
    Last Post: August 2nd 2010, 06:03 AM
  2. Complex Number--De Moivres Theorm
    Posted in the Pre-Calculus Forum
    Replies: 6
    Last Post: July 29th 2010, 03:26 AM
  3. Residue Theorm - Q1
    Posted in the Differential Geometry Forum
    Replies: 6
    Last Post: October 3rd 2009, 08:35 AM
  4. Remainder Theorm
    Posted in the Algebra Forum
    Replies: 7
    Last Post: December 16th 2007, 09:33 AM
  5. Green's Theorm
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 27th 2006, 01:28 AM

Search Tags


/mathhelpforum @mathhelpforum