Cauchy's theorem

Cauchy's theorem, also known as Cauchy's integral theorem, is a theorem in complex analysis which states that any line integrals sharing the same endpoints, if the region between the paths is holomorphic, will have the same value. This means that when taking the integral of a holomorphic function between two points in the complex plane, the path between them need not be specified, and will be equal to

$\int\limits_{z_1}^{z_2}f(z)dz=F(z_2)-F(z_1)$

As a corollary, any closed line integral over a holomorphic function will have a value of zero. It can be easily proved from this property that if there are two closed line integrals, one around the other, and the region between them is holomorphic, they will have the same value. The converse of Cauchy's theorem, known as Morera's theorem, states that if every closed line integral over a function is equal to zero, the function is holomorphic.