what is the cauchy-coursat theorem state specificly in word?
Is it calculate the area or the legth,which area and what legth?
please give a example!
And what is deformation of contour?
It's Cauchy-Goursat by the way. Anyways, it says that if you integrate a function on a rectifiable curve which is contained within some simply connected region for which your function is holomorphic then the integral is zero. By deformation what is really meant is a 'homotopy' (google it). You can use CG to prove that if $\displaystyle \Gamma$ and $\displaystyle \Gamma'$ are curves for which there exists a homotopy which doesn't pass through a singularity (intuitively) then $\displaystyle \displaystyle \int_\Gamma f\;dz=\int_{\Gamma'}f\;dz$.
Hope this helps:
Cauchy's integral theorem - Wikipedia, the free encyclopedia
Edited: Sorry, I didn't see Drexel28's post.