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Residues and the fundamental group

by alexfloo
Tags: fundamental, residues
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alexfloo
#1
Mar24-12, 08:23 PM
P: 192
I've been thinking about complex residues and how they relate to the topology of a function's Riemann's surface. My conclusion is this: it definitely tells us something, but it relates more directly to the Riemann surface of its antiderivative. Specifically:

A closed contour in the plane is closed when projected to the Riemann surface of f's antiderivative iff the sum of the residues of f interior to it are zero.

Is this correct?
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DonAntonio
#2
Mar30-12, 02:16 PM
P: 606
Quote Quote by alexfloo View Post
I've been thinking about complex residues and how they relate to the topology of a function's Riemann's surface. My conclusion is this: it definitely tells us something, but it relates more directly to the Riemann surface of its antiderivative. Specifically:

A closed contour in the plane is closed when projected to the Riemann surface of f's antiderivative iff the sum of the residues of f interior to it are zero.

Is this correct?

What do you mean mathematically by "projected to the Riemann Suerface of f's derivative"? What kind of projection are you thinking about? Can you give some example(s)?

DonAntonio


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