the Cauchy-Goursat theorem
/ Goursat: goor-SAH /
Cauchy's original proof of his integral theorem quietly assumed the derivative f' was continuous, because it leaned on Green's theorem. Goursat's achievement was to remove that crutch entirely. The Cauchy-Goursat theorem is the same conclusion — the integral of a holomorphic f around a closed contour in a simply connected domain is zero — but proved under the leanest possible hypothesis: f need only be complex-differentiable, with no assumption that f' is continuous.
Why this is a big deal: a priori, complex differentiability is just the existence of a single limit at each point, a seemingly mild requirement. Goursat showed it already forces the integral theorem. And once the integral theorem holds, the Cauchy integral formula follows, and from that formula one proves that a holomorphic function is automatically infinitely differentiable and equals its Taylor series. So the continuity of f' that Cauchy assumed turns out to be a free consequence, not a needed input. Goursat's careful bookkeeping is what lets the theory bootstrap from the weakest assumption to the strongest conclusions.
The proof technique is the triangle argument: prove the theorem first for the boundary of a triangle by repeated subdivision, then extend to polygons by cutting them into triangles, and finally to arbitrary contours by approximation. Establishing it for triangles is the crux, because every later step reduces to that base case. The payoff is foundational honesty — the miracle of complex analysis rests on differentiability alone, exactly as one would hope.
Take f holomorphic on a disk and a triangle T inside it. Subdivide T into four congruent sub-triangles by joining edge midpoints; the integral over T's boundary is the sum over the four sub-boundaries (shared inner edges cancel). At least one sub-triangle carries an integral at least one quarter of the whole. Iterate, zooming in on a point where differentiability makes the integral negligibly small, and conclude the original integral is 0.
The quadrisection step at the heart of Goursat's triangle proof.
Cauchy-Goursat and 'Cauchy's integral theorem' name the same conclusion; the hyphenated form simply credits Goursat for proving it without assuming f' is continuous.