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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.
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 )?
cgi-bin 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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