for every closed curve C in D, then f must be holomorphic on D.
The assumption of Morera's theorem is equivalent to that f has an anti-derivative on D.
The converse of the theorem is not true in general. A holomorphic function need not possess an antiderivative on its domain, unless one imposes additional assumptions. For instance, Cauchy's integral theorem states that the line integral of a holomorphic function along a closed curve is zero, provided that the domain of the function is simply connected.
There is a relatively elementary proof of the theorem. One constructs an anti-derivative for f explicitly. The theorem then follows from the fact that holomorphic functions are analytic.
Without loss of generality, it can be assumed that D is connected. Fix a point a in D, and define a complex-valued function F on D by
The integral above may be taken over any path in D from a to b. The function F is well-defined because, by hypothesis, the integral of f along any two curves from a to b must be equal. It follows from the fundamental theorem of calculus that the derivative of F is f:
In particular, the function F is holomorphic. Then f must be holomorphic as well, being the derivative of a holomorphic function.
for every n, along any closed curve C in the disc. Then the uniform convergence implies that
for every closed curve C, and therefore by Morera's theorem f must be holomorphic. This fact can be used to show that, for any open set Ω ⊆ C, the set A(Ω) of all bounded, analytic functions u : Ω → C is a Banach space with respect to the supremum norm.
or the Gamma function
to be zero for every closed triangle T contained in the region D. This in fact characterizes holomorphy, i.e. f is holomorphic on D if and only if the above conditions hold.