Ignoring the 1st part of the 1st sentence of the question all I want to get is a branch $f$ of the inverse function of $g(z)=z^4. $ This is how I set about doing it, however, I need to verify this.
Let $f(z)=e^{\frac{L_0(z)}{4}}$ where $L_0:\mathbb{C}$\ $[0,\infty)\rightarrow \mathbb{C}$ is such that $L_0(z)=log|z|+i\theta(z)$ and $0<\theta(z)<2\pi$.
It follows that $$g(f(z))=[e^{\frac{L_0(z)}{4}}]^4=[e^{L_0(z)}]=z $$ Since $f$ is continuous on D $f$ is a branch of $g^{-1}$ on D. Also it can be easily find $f(-1)$. I need to know whether all my steps are correct and also need assistance with finding the other branches. Thanks (Source of the exercise: Palka)
To find the other branches, just consider this question: If you are given one fourth root $w$ of a number $z$, how do you find the other fourth roots of $z$ in terms of $w$?