How to find the inverse of the complex function $ f(z)=\dfrac{z}{\sqrt{1+|z|^{2}}}$

Question

I'm trying to calculate the inverse function of : $$ f(z)=\dfrac{z}{\sqrt{1+|z|^{2}}}$$ where $z\in$ $\mathbb{C}$.

Can someone help me?

Answer 1

If you mean the inverse, $1-|f|^2=\frac{1}{1+|z|^2}\implies z=f\sqrt{1+|z|^2}=\frac{f}{\sqrt{1-|f|^2}}$.