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