Show that $\log|\sin(z)|$ is the real part of a holomorphic function

Question

$D$ is a connected, simply connected domain with $\sin(z)$ never zero on D. Show that $\log|\sin(z)|$ is the real part of a holomorphic function.

My question is: how to show $\sin(z)$ maps a simply connected set to a simply connected set?

Answer 1

My question is: how to show $\sin(z)$ maps a simply connected set to a simply connected set?

It need not. If $D$ is for example the upper half plane, the line $\operatorname{Im} z = 1$ is mapped to an ellipse that winds around $0$. However, since $D$ is by assumption simply connected, and $\sin$ has no zeros in $D$, the logarithmic derivative

$$\cot z = \frac{\cos z}{\sin z}$$

has a primitive $h$ on $D$. $h'(z) = \cot z$ means that $e^{-h(z)}\sin z$ is constant, and by adding a suitable constant to $h$ you can achieve

$$\sin z = e^{h(z)}$$

on $D$. Then you have $\log \lvert\sin z\rvert = \operatorname{Re} h(z)$.

Answer 2

Try $$\log(sin(z)).$$ Recall that $\log(re^{i\theta}):=\log(|r|)+i(\theta+2\pi k)$.