Homology and Integration; Req. for Proof.

[Bacle]

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.

 

[lavinia]

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.

What is C^1(D)?

 

[Bacle]

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.

 

[lavinia]

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.

Integrate
f(x,y) = xy^{2}

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?

 

[Bacle]

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 ?

 

[lavinia]

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 ?

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.