Problem 1594 (difficulty: 6/10)

Let \(\displaystyle D\) be a simply connected domain and \(\displaystyle f(z)\) a non-vanishing holomorphic function on \(\displaystyle D\).

(a) Show that \(\displaystyle f'(z)/f(z)\) has an antiderivative on \(\displaystyle D\).

(b) Show that if \(\displaystyle g'=f'/f\) on \(\displaystyle D\), then \(\displaystyle f(z)e^{-g(z)}\) is constant on \(\displaystyle D\).

(c) Show that \(\displaystyle \log f\) has a continuous branch on \(\displaystyle D\).

Give me another random problem!