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I 'm trying to prove the integral$\frac{1}{2\pi i}\int_{C}\frac{dz}{z}$is an integer,where $C:[0,1]\to S^1\subset \mathbb{C}$ is a continuous function and $C(0)=C(1)=1$.

I know the proof when $C$ is $C^1$,but here $C$ is just continuous(maybe not bounded variation).

There are 2 questions:

  1. How to define $\int_{C}\frac{dz}{z}$ when $C$ is not differentiable? Use Riemann-Stieltjes integral?(and does this integral always exist?)
  2. How to prove $\frac{1}{2\pi i}\int_{C}\frac{dz}{z}\in\mathbb{Z}$?
Isllier
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  • Is this what you are asking? https://math.stackexchange.com/questions/1820326/when-is-the-line-integral-independent-of-parameterization – b00n heT May 27 '24 at 15:16
  • That question talks about the case when$C$ is smooth,but here $C$ is just continuous. – Isllier May 27 '24 at 15:30
  • Is there a factor of $2\pi i$ missing? – Cameron L. Williams May 27 '24 at 15:39
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    I remember that Conway's complex analysis book has something about it. For your first question, the integral is defined by Riemann-Stieljes integral and the contour need to be rectifiable; for your second question, I think he used Leibniz's law. – Zhang May 27 '24 at 15:51
  • Here's a related question(https://math.stackexchange.com/questions/2559707/how-to-show-that-the-integral-frac12-pi-i-int-gamma-fracdzz-a?rq=1), but I still want a example that this integral not exists for $C$ not rectifiable. – Isllier May 27 '24 at 17:02

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One way of doing the first part of your question is in Lang's book on complex analysis. More generally suppose that $f$ is holomorphic on an open set $U$. For every continuous curve $C:[0,1]\to U$ and $z$ in the image $C([0,1])$ choose a small enough open ball $B_z$ centered at $z$ and such that $f$ has a primitive here. A compactness argument gives us a partition $0=t_0<t_1<...<t_{n-1}<t_n=1$ and finitely many of the open balls $B_z$ such that $C([t_k,t_{k+1}])\subset B_k$ for $k=0,...,n-1$.

Now simply define $$\int_Cf(z)dz=\sum_{k=0}^{n-1}F_k(C(t_{k+1}))-F_k(C(t_k))$$ where $F_k$ is a primitive of $f$ on $B_k$. It is not too hard to check that this definition is actually independent of the choice of partition, open balls and primitives.

spinosarus123
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