Let \(\Omega \subseteq \mathbb{C}\) be a connected open region. (1) Prove that if \(f: \Omega \to \mathbb{C}\) is a non-constant holomorphic function, then \(f'\) is holomorphic on \(\Omega\) and the zeros of \(f'\) are isolated in \(\Omega\). (2) Prove that if \(f, g: \Omega \to \mathbb{C}\) are holomorphic functions satisfying \(f'(z) = g'(z)\) for all \(z \in \Omega\), then \(f(z) - g(z) = C\) for some constant \(C \in \mathbb{C}\).
Solution
Because \(f\) is holomorphic, \(f'\) is holomorphic.
If the zeros of \(f'\) had an accumulation point in \(\Omega\), the identity theorem would give \[ f'\equiv0. \] A holomorphic function with zero derivative on a connected domain is constant, contradicting the hypothesis. Hence the zeros of \(f'\) are isolated.
For the second statement, let \[ h=f-g. \] Then \(h\) is holomorphic and \[ h'=f'-g'=0. \] Therefore \(h\) is constant on the connected domain \(\Omega\). Thus \[ f-g=C \] for some \(C\in\mathbb C\).