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cgibin science forum beginner
Joined: 19 May 2005
Posts: 6

Posted: Tue Jun 27, 2006 3:08 am Post subject:
Line Integrals On a Riemann Surface



I am interested in defining and evaluating line integrals on a Riemann
surface and evaluating them using residues. For example, given the
Riemann surface
M= { (z,w)\in C^2  w=z^2 },
the closed contour
L={ (e^{it/2},e^{it}) \in C^2  t \in R } \subset M,
and a function f1:M > C which is an extension of a function f0,1) >
R.
I want to define the integral of f over L. I also would like to
evaluate this integral using residues.
I have looked at various books on Riemann Surfaces, Serveral Complex
Variables, but I have no readable and relevant sources.
Does anyone have a good reference? Can anyone provide a brief rundown
of the necessary machinery. 

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lucky.math science forum beginner
Joined: 27 Jun 2006
Posts: 1

Posted: Tue Jun 27, 2006 7:33 am Post subject:
Re: Line Integrals On a Riemann Surface



I think what you need is not integral on Riemann surface,but on line.
Maybe ,you can just integral it on (0,1 )?
cgibin wrote:
Quote:  I am interested in defining and evaluating line integrals on a Riemann
surface and evaluating them using residues. For example, given the
Riemann surface
M= { (z,w)\in C^2  w=z^2 },
the closed contour
L={ (e^{it/2},e^{it}) \in C^2  t \in R } \subset M,
and a function f1:M > C which is an extension of a function f0,1) 
R.
I want to define the integral of f over L. I also would like to
evaluate this integral using residues.
I have looked at various books on Riemann Surfaces, Serveral Complex
Variables, but I have no readable and relevant sources.
Does anyone have a good reference? Can anyone provide a brief rundown
of the necessary machinery. 


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