Results 1 to 3 of 3

Math Help - Riemmann Surface integral

  1. #1
    Bop
    Bop is offline
    Junior Member
    Joined
    Dec 2009
    Posts
    48

    Riemmann Surface integral

    Hello, here's the problem:

    Let be C a curve in Riemmann Surface log(z) that goes from z=1 (in the log(1)=0 sheet) to z=1 turning around z=0 'n' times. Calculate:


    \int_C  log(z) dz


    I have tried to calculate in this way:

    log(z)=log(r)+it

    z(t)=r(cos(t)+isin(t)) so d(z(t))=r(-sin(t)+icos(t))dt

    thus:

    \int_C  log(z) dz=\int^{2n\pi}_0 logz(t)z'(t)dt=\int^{2n\pi}_0(log(r)+it)r(-sin(t)+icos(t))dt=2\pi i rn

    As r=1: (I know I could have simplified before)

    \int_C  log(z) dz=2\pi i n


    What do you think? Is it right?


    Thank you.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Bop
    Bop is offline
    Junior Member
    Joined
    Dec 2009
    Posts
    48
    I'm not sure about the 'n'..

    Thank you.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Aug 2008
    Posts
    903
    \int_{C(n)} \log(z)dz=2n\pi i

    Everytime you go one time around, the integral is 2\pi i. You could show this piece-wise for each circuit, or I would analytically extend the antiderivative and write:

    \int_{C(n)}\log(z)dz=\left(z\log(z)-z\right)\biggr|_1^1=(2n\pi i-1)-(0-1)=2n\pi i

    it's meant to look controversial just to get people thinking

    I do believe an argument for it's validity could be made however.
    Last edited by shawsend; May 20th 2010 at 11:46 AM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Surface Integral
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 5th 2011, 06:43 AM
  2. Surface Integral
    Posted in the Calculus Forum
    Replies: 8
    Last Post: May 19th 2011, 06:50 AM
  3. Surface Integral
    Posted in the Calculus Forum
    Replies: 0
    Last Post: April 3rd 2011, 05:28 AM
  4. Volume integral to a surface integral ... divergence theorem
    Posted in the Advanced Applied Math Forum
    Replies: 3
    Last Post: October 7th 2010, 07:11 PM
  5. Surface Integral
    Posted in the Calculus Forum
    Replies: 5
    Last Post: July 31st 2008, 03:26 AM

Search Tags


/mathhelpforum @mathhelpforum