Q: Find all functions that are analytic and satisfy $(\operatorname{Re}(f(z)))^2 = \operatorname{Im}(f(z))$ I've been trying to find all functions $f(z):\Bbb{C}\longrightarrow\Bbb{C}$ that are analytic and satisfy
$$(\operatorname{Re}(f(z)))^2 = \operatorname{Im}(f(z))$$
After plugging in $f(z)=x+iy$, I got $f(z)=x+ix^2$. 
These kind of functions never satisfy Cauchy-Riemann equations, since $\mathcal u_x=1 \neq 0=v_y$. 
Does it follow from this argumentation that no such analytic $f(z)$ exists?

Another kind of solution: Note that ${\rm Im}(f(z))\geq 0$ for all $z$. Hence if we put $g(z)=\exp(if(z))$, we have $|g(z)|=\exp(-{\rm Im}(f(z))\leq 1$ for all $z$. By Liouville's theorem, $g$ is constant, and $f$ is also a constant, of the form $c+ic^2$, $c\in \mathbb{R}$. 
