Consider $\overline{R}:[-a,a]\times [-b,b]$ and $\varphi: [-a,a] \rightarrow [-b,b]$ a continous map. Let $K$ be the graphic of $\varphi$. if $f: int(\overline{R}) \rightarrow \mathbb{C}$ is continous and holomorphic in $int(\overline{R})\K$, show that $\int_{\partial R_0} f(z)dz=0$ for all rectangle $R_0$ strictly contained in $int(\overline{R})$.
It seems to be the proof of Cauchy-Goursat Theorem for multiply convex domains (I think that is how is written in English) and I know that this proof consists in creating small rectangles $R_1,R_2, ..., R_n$ around each singularity, such that all of them is containd in $R_0$ and $\bigcap_{i=1}^n R_i = \emptyset$, however, since these singularities are in $\varphi \cap \partial R_0 \subset \partial R_0$, I do not have idea to create these rectangles.
Am I taking the wrong path? Could you suggest another one, if I'm wrong? I'm gonna insert an image of what I am trying to say, if it is not clear.
