I want a $C^\infty$ function $f:U\rightarrow\mathbb{C}$, where $U\subset\mathbb{C}$ is a neighborhood of $0$, such that there exists a sequence of points $z_n\in U-\{0\}$ and $\lim_{n\rightarrow\infty} z_n=0$, satisfy $(f(z_n))^2=z_n^3$.
I faced this problem in my research. I want to find a smooth substite for $z^{3/2}$, since $z^{3/2}$ cannot be defined as a single-value function and has bad behavior at $0$ (only $C^{1,1/2}$, not smooth).
By inverse function theorem, if $f$ satisfies the condition, then the Jacobian of $f$ at $0$ must be degenerate. It's quite difficult for me to really find a $f$, since it seems very hard for a real-analytic $f$ to satisfy those conditions, besides I have limited knowledge about non-real-analytic $C^\infty$ functions.
Any comments and answers are welcome. You can also replace $3/2$ by other half-integers, like $5/2$, $7/2$, etc. I'll really appreciate your help.
Partial Answer
It is easy to see that a real Analytic function cannot exists. Indeed if $f$ is analytic, then $f^2$ is also analytic and the set $\{ f^2(z)=z^3 \}$ has an accumulation point. Then it would follow that $f^2(z)=z^3$.
Same argument can be made about any other half integer.