Morera's theorem
In complex analysis, a branch of mathematics, Morera's theorem, named after Giacinto Morera, gives a criterion for proving that a function is holomorphic. Morera's theorem states that a continuous, complex-valued function f defined on an open set D in the complex plane that satisfies ∮ γ f ( z ) d z = 0 {\displaystyle \oint _{\gamma }f(z)\,dz=0} for every closed piecewise C1 curve γ {\displaystyle \gamma } in D must be holomorphic on D. The assumption of Morera's theorem is equivalent to f having an antiderivative on D. The converse of the theorem is not true in general.