Complex analysis

  • Thread starter sbashrawi
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  • #1
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Homework Statement



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

Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
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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.
 
  • #3
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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
 
  • #4
1,838
7
Take the real part of the integral expression by adding the complex conjugate.
 
  • #5
55
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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
 
  • #6
1,838
7
integr( f + conj(f))*f'dz =

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

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

2i integr u dv
 
  • #7
55
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Thank you very much
 

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