Cauchy-Riemann equations for a holomorphic function given in polar form

Question

Suppose $f(re^{i\theta})=R(r,\theta)e^{i\Theta(r,\theta)}$ describes a holomorphic function. Is there a Cauchy-Riemann system that $R(r,\theta)$ and $\Theta(r,\theta)$ must satisfy? I've seen this question answered for the related case $f(re^{i\theta})=U(r,\theta)+iV(r,\theta)$ but not when $f$ is given in polar form. One way to proceed here is to write $$f(re^{i\theta})=\underbrace{R(r,\theta)\cos\big(\Theta(r,\theta)\big)}_{U(r,\theta)}+i\underbrace{R(r,\theta)\sin\big(\Theta(r,\theta)\big)}_{V(r,\theta)}$$ and then use the Cauchy-Riemann equations \begin{equation}\label{eq:1}\frac{\partial U}{\partial r}=\frac{1}{r}\frac{\partial V}{\partial\theta}~\text{ and }~\frac{\partial V}{\partial r}=-\frac{1}{r}\frac{\partial U}{\partial\theta}\end{equation} which were alluded to above. This approach doesn't seem to lead to a nice first order system comparable to the Cauchy-Riemann equations, though. Perhaps there a simplification I'm missing? Has someone seen anything like this in the literature?

Answer 1

If $z\mapsto f(z)=R(z)e^{i\Theta(z)}$ is analytic and nonzero in a simply connected domain $\Omega$ not containing the origin then $f$ has an analytic logarithm $g:\Omega\to{\mathbb C}$, meaning that $e^{g(z)}=f(z)$ in $\Omega$. One then has $$g(z)=\log R(z)+i\Theta(z)\ ,\tag{1}$$ up to an additive constant $2k\pi i$. If $z$ is expressed in polar coordinates: $z=re^{i\theta}$, then the CR equations $$U_r={1\over r}V_\theta,\qquad V_r=-{1\over r}U_\theta\ ,$$ applied to $(1)$, translate into $${R_r\over R}={1\over r}\Theta_\theta,\qquad \Theta_r=-{1\over r}{R_\theta\over R}\ .$$