Cauchy's integral theorem
/ Cauchy: koh-SHEE /
This is the theorem the whole subject pivots on, and it is startlingly clean. If f is holomorphic on a simply connected domain (a region with no holes), then the integral of f around every closed contour in that domain is exactly zero. No computation, no special functions — just zero, every time. From this single fact almost all of complex analysis unfolds.
Read it through the chapter's equivalence and it says even more. Because closed-loop integrals all vanish, f's integrals are path-independent on the domain, and f possesses a primitive there. So Cauchy's theorem is really the statement that every holomorphic function on a hole-free region has an antiderivative — the strongest possible good behavior. The hypotheses earn their keep: 'holomorphic' (complex-differentiable) is the analytic input, and 'simply connected' is the topological input ensuring there is no hole for a loop to trap a singularity around.
The honest boundaries matter. Drop simple connectivity and the theorem can fail: 1/z is holomorphic on the punctured plane, yet its integral around a loop circling the origin is 2 pi i, because the puncture is a hole the loop encloses. Drop holomorphy at even one point inside the loop and it can fail too. What is remarkable, and the content of the Cauchy-Goursat sharpening, is that mere complex-differentiability suffices — you do not need to assume the derivative is continuous, even though the conclusion will retroactively prove f is infinitely differentiable.
On the disk |z| < 2 (which is simply connected), e^z, cos z, and any polynomial are holomorphic, so the integral of each around the circle |z| = 1 is 0. But 1/z fails the hypothesis on any region containing 0, and indeed its integral around |z| = 1 is 2 pi i, not 0.
Cauchy's theorem: a hole-free domain plus holomorphy forces every closed-loop integral to zero.
It does not say a closed-loop integral is always zero — it says so only when f is holomorphic everywhere inside, including on the part enclosed by the loop; a singularity caught inside changes the answer entirely.