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 and are curves for which there exists a homotopy which doesn't pass through a singularity (intuitively) then .