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Homework Help: Complex analysis

  1. Mar 28, 2010 #1
    1. The problem statement, all variables and given/known data

    Let gama be a closed curve and f be analytic function. Show that the integration of f(z)f' dz is puerly imaginary

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Mar 30, 2010 #2
    Welcome sbashrawi,
    As f(z)f'(z) = (1/2) d/dz f^2 , Cauchy's formula shows that what you claim is invalid unless gamma encircles some poles of f with real residues at them.
     
  4. Mar 30, 2010 #3
    Thank you very much

    I am sorry the true statment is that :

    integration of ( conjugate of f ) * f' *dz is purely imaginary.

    I tried to prove it using the winding number but I couldn't
     
  5. Mar 30, 2010 #4
    Take the real part of the integral expression by adding the complex conjugate.
     
  6. Mar 30, 2010 #5
    Hi

    here is what I did:

    integ(conj(f)*f'dz) = integr( f + conj(f))*f'dz
    which implies

    integ [ conj(f) - 2 Re(f)] * f' dz = 0

    letting f = u + iv , then the expression will be
    integ[ -u -iv] * f' dz = 0

    then I couldn't find how it is purely imaginary from this step
     
  7. Mar 31, 2010 #6
    integr( f + conj(f))*f'dz =

    integr (2Re(f))*f'dz =

    integr[(2 u) (du + i dv)] =

    2i integr u dv
     
  8. Mar 31, 2010 #7
    Thank you very much
     
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