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Homology and Integration; Req. for Proof.

  1. Jul 23, 2011 #1
    Hi, All:
    Let D be a domain (open+ connected) in the complex plane.

    I was wondering if someone had a ref. for the proof that if two
    curves C,C' in D are homotopic, thenfor f in C^1(D), Int_C F= Int_C' f, and/or if this
    theorem has a standard name.

    Thanks.
     
  2. jcsd
  3. Jul 23, 2011 #2

    lavinia

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    What is C^1(D)?
     
  4. Jul 24, 2011 #3
    I meant f is continuously-differentiable in D. This seems like it may be
    a corollary of Green's thm., but I am not sure.
     
  5. Jul 24, 2011 #4

    lavinia

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    Integrate
    f(x,y) = xy[itex]^{2}[/itex]

    along the edges of the unit square.

    Along the edges (0,0) to (1,0) and (0,0) to (0,1) the integral is zero.
    Along the edge (1,0) to (1,1) the integral is 1/3.
    From (0,1) to (1,1) the integral is 1.

    Did you mean the integral of df?
     
  6. Jul 24, 2011 #5
    You're right, I think I do not have the conditions right. I was thinking of a def. of

    homology of SCCurves, as defined in Rotman's Homological Algebra book

    ( I don't have the book with me at the moment; not in my school's library):

    We are working in a subset S of R^2 , and f is defined there ; I am

    not sure if f is assumed (complex) analytic,but then two simple-closed curves C,C' in S are homologous

    if (Def.) Int_C f= Int_C' f , so that Int_(C-C') f = Int_0 f , so I cannot remember

    the actual conditions on f. But basically the result is that the integral of f is constant

    in homology.

    Still, any chance you have a ref. for Int dz ?
     
    Last edited: Jul 24, 2011
  7. Jul 24, 2011 #6

    lavinia

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    The theorem is if a 1-form is closed then its integral on homotopic curves will be the same. If f is holomorphic in the domain then the 1 form fdz is closed. This follows from the Cauchy-Riemann equations. (For complex 1-forms closed means that the real and imaginary parts are both closed.)

    The general theorem is Stokes theorem in the plane which classically I think is Green's theorem.
     
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