In complex analysis, a branch of mathematics, Morera’s theorem, named after Giacinto Morera, gives an important criterion for proving that a function is holomorphic.
Morera’s theorem states that if is a continuous, complex-valued function defined on an open set in the complex plane, satisfying
for every triangle in , then must be holomorphic on .
The assumption of Morera’s theorem is equivalent to that has an anti-derivative on . 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.
Now we state and prove the following generalization of the Morera’s theorem: Suppose that is continuous on , and
for every circle . Prove is holomorphic.
Proof. If is a smooth bounded function with
so is an approximate identity.
that is, the convolution of and . Then by Folland Prop. 8.10 is smooth (actually smooth on compact subsets which implies smooth everywhere). By Folland Thm. 8.14 uniformly on compact subset as and
By a change of variables
where is the translate of by which is still a circle. So
Thus if the partial derivatives all exist and are continuous and
Since this is true for any circle (any translate or dilate) this implies
the C-R equations for so is holomorphic and since it converges uniformly on compact subsets to , is also holomorphic.