7.2 Morera’s Theorem
Let \(f(z)\) be a continuous function in a simply connected domain \(D\) and such that
\[\int _C f(z)dz = 0\]
for every simple closed curve \(C\) in \(D\). Then \(f(z)\) is analytic in \(D\).
Proof. We have that if the points \(z_0\) and \(z\) are within \(D\), then the function satisfies the conditions of the theorem \[F(z) = \int ^z_{z_0} f(\xi ) d\xi \] is analytic such that \(\, F'(z) = f(z)\). Thus \(f(z)\) is seen to be the derivative of the analytic function \(F(z)\) which itself is analytic. □
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.