Let $W=\{\operatorname{Re} z>0\}$. I want to find all analytic functions $f: W \rightarrow \mathbb{C}$ for which there exists a $C^\infty$ function $\phi: (0,\infty)\rightarrow \mathbb{R}$ such that $\operatorname{Re}(f(z))=\phi(|z|^2)$.
I feel like it should be something like $\log z^2$, but I don't know a general approach to find all of such functions.
For $f(z)=u+iv$ to be analytic, it satisfies the Cauchy-Riemann equations(in polar form): $$\frac{\partial u}{\partial r}=\frac1r \frac{\partial v}{\partial \theta} $$ $$\frac1r \frac{\partial u}{\partial \theta} =-\frac{\partial v}{\partial r} $$
Thus, we obtain $$\frac{\partial v}{\partial \theta}= r\frac{d\phi(r^2)}{dr}$$ $$\frac{\partial v}{\partial r}=0$$
From the first equation, we obtain $$v= r\theta\frac{d\phi(r^2)}{dr}+C_1$$($C_1$ is real.)
However, from the second equation, $v$ has no dependence on $r$. Therefore, $$r\frac{d\phi(r^2)}{dr} =k$$($k$ is real constant.)
Let $r=\sqrt R$, $$\sqrt R\frac{d\phi(R)}{dR}\frac{dR}{d\sqrt R}=k$$ which gives $$\phi(r^2)=k\ln r+C_2$$
Thus, $$f(z)=k\ln r+ik\theta+C=k\ln(z)+C$$ for any complex number $C=C_2+iC_1$.