# cauchy theorem over cycles homologous to zero

Definitions:

$i)$ A cycle $\gamma$ is a finite sequence of continuous oriented closed paths in the complex plane. We denote $\gamma = (\gamma_1,...\gamma_n)$ where $\gamma_k$ are the closed paths of the cycle. We define $\int\limits_\gamma {f\left( z \right)dz = \sum\limits_{k = 1}^n {\int\limits_{\gamma _k } {f\left( z \right)dz} } }$.

$ii)$ The index of a point $c$ with respect to the cycle $\gamma=(\gamma_1,...\gamma_n)$ , is $I(\gamma,c) = I(\gamma_1,c)+...+I(\gamma_n,c)$.

$iii)$ A cycle with range contained in a domain (open and connected) $U\subset \mathbb{C}$ is said to be homologous to zero with respect to U , if $I(\gamma,c)=0$ for every $c \in \mathbb{C}-U$.

Well sorry for all these definitions. But I have a question with Cauchy theorem , but involving cycles.

The classical version is this: Cauchy theorem: If f is holomorphic on an open set D ( then $f(z)dz$ is a closed form by morera theorem) and $\gamma$ is a continuous closed path in D , that is homotopic to a point in D , then $\int _{\gamma} f(z)dz=0$

My question:

$i)$ Cauchy theorem 2. If if is an analytic in a domain $U\subset \mathbb{C}$ , then: $\int\limits_\gamma {f\left( z \right)dz = 0}$ for every cycle $\gamma$ that is homologous to zero in U.

Well.. I don't know how to prove this, is likely to be very simple, but I don't know how to prove it here, because in the last case I could define an homotopy, and concluding the result, but here I can't :S

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What I would like to know now, is what $I(\gamma_i, c)$ is... Once that is clear, you can probably show that any $c\in \Bbb C-U$ must be outside any null-homological $\gamma$ contained in $U$, and thus $\gamma$ null-homotopic in $U$. –  Arthur Oct 19 '12 at 22:41
$I(\gamma_i,c)$ is the winding number of the point $c$ in the curve $\gamma_i$ $$I(\gamma_i,c)=\int _{\gamma_i} \frac{dz}{z-c}$$ –  Daniel Oct 19 '12 at 23:56