Cauchy's integral theorem

Cauchy's integral theorem is the foundational miracle of the subject: if a function is holomorphic everywhere inside and on a closed loop, then integrating it once around that loop gives exactly zero. The walk around the boundary cancels itself out completely. The intuition is that a holomorphic integrand has, locally, an antiderivative, and the integral of a derivative around any closed path returns you to where you started — net change zero.

A clean statement: if f is holomorphic on a simply connected open set U (a region with no holes), then for every closed contour gamma lying in U, the contour integral of f over gamma is 0. The simple-connectivity hypothesis is essential — it guarantees there is no hidden singularity for the loop to encircle. Goursat's sharpening shows you need only assume f is complex differentiable, not that the derivative is continuous, which is why it is sometimes called the Cauchy–Goursat theorem.

The theorem immediately gives path independence: between two endpoints, the integral of a holomorphic function does not depend on the route taken (within a simply connected region). It is also the springboard for the integral formula and the residue theorem. The crucial caveat is the topology: over a region WITH a hole, or a loop that does encircle a singularity, the integral can be nonzero — the integral of 1/z around the unit circle is 2pi i precisely because the origin is a hole the loop wraps around.