The problem is to describe all branches and all the curves of analytic continuation of $\sqrt{4z-\sqrt[3]{z}}$.
I started with representing function in a way $\sqrt{4w^3-w}\circ\sqrt[3]{z}$ So, there are 6 branches:
- $\sqrt{4w^3-w}_{(0)}\circ\sqrt[3]{z}_{(0)}$
- $\sqrt{4w^3-w}_{(0)}\circ(-\frac{1}{2}-i\frac{\sqrt{3}}{2})\sqrt[3]{z}_{(0)}$
- $\sqrt{4w^3-w}_{(0)}\circ(-\frac{1}{2}+i\frac{\sqrt{3}}{2})\sqrt[3]{z}_{(0)}$
- $-\sqrt{4w^3-w}_{(0)}\circ\sqrt[3]{z}_{(0)}$
- $-\sqrt{4w^3-w}_{(0)}\circ(-\frac{1}{2}-i\frac{\sqrt{3}}{2})\sqrt[3]{z}_{(0)}$
- $-\sqrt{4w^3-w}_{(0)}\circ(-\frac{1}{2}+i\frac{\sqrt{3}}{2})\sqrt[3]{z}_{(0)}$
Now we need to find out where continuation exists. Since $\sqrt{4w^3-w}$ is not analytic in 0, we should exclude $w=0, \pm\frac{1}{2}$. This means we have to exclude 0 from all branches, exclude $\pm\frac{1}{8}$ from branches 1,4; $\pm\frac{1}{8}\frac{1}{(-\frac{1}{2}-i\frac{\sqrt{3}}{2})^3}$ from 2,5; $\pm\frac{1}{8}\frac{1}{(-\frac{1}{2}+i\frac{\sqrt{3}}{2})^3}$ from 3 and 6. If we have a curve $\gamma$, area of continuation around point $\gamma(t)$ for branches 1 and 4 will be a ball $B(\gamma(t), min(|\gamma(t)|,|\gamma(t)-\frac{1}{8}|,|\gamma(t)+\frac{1}{8}|))$ and analog.
Let's see what is the value of continued function is on the curve $\gamma$. Let's assume $l(z)=ln(|z|)+iarg(z)$ So there should be something like $f(z)=e^{\frac{1}{2}l(e^{\frac{1}{3}l(z)})}$, but I don't understand if it's right or should be done other way.
My questions are:
- Are my thouhts on the problem true or I've chosen absolutely wrong way?
- How exactly should be written $f(z)$ on the curves of continuation? Or, at least, what can I read to have an enlightenment there.