Cauchy-Goursat theorem

Goursat proved the theorem without assuming that the derivative \(f'\) is continuous, only that \(f\) is differentiable at every point; Cauchy had proved it earlier with that extra hypothesis. The idea of the proof is to cut the region into small triangles: on each one the function looks like a linear function, whose closed integral is zero, and the contributions of the interior edges cancel.

The hypothesis is twofold: \(f\) must be holomorphic at every point of the region enclosed by the curve, not just on it, and that region cannot have holes. That is why \(\oint_{|z|=1}\frac{dz}{z}=2\pi i\neq 0\): the function \(1/z\) is holomorphic on the circle, but not at the point \(z=0\) inside it.

An immediate consequence is contour deformation: if \(f\) is holomorphic on the region between two closed curves, both integrals agree, and a curve can be shrunk or stretched without changing the value as long as it does not cross a problem point. It also implies that the integral between two points does not depend on the path, which leads to the primitive.

A typical mistake is reading the theorem backwards: a zero integral does not prove that \(f\) is holomorphic. For example, \(\oint_{|z|=1}\frac{dz}{z^2}=0\) even though \(1/z^2\) is not holomorphic at the origin. The theorem gives a sufficient condition, not a necessary one.