Considering $X$ be the Riemann surface of ananalytic function $\sqrt{\sqrt{z − 1} − 2}$, I would like to know how many points of $X$ there are over $z = 1$, and $z = 5$? Then I'm asked to choose a local coordinate in all these points and expand that analytic function in power series with respect to that parameter.
I believe there are only 2 points in $z = 1$: $z_1 = (1, -i\sqrt{2})$ and $z_2 = (1, i\sqrt{2})$; and two points in $z = 5$: $(5,0)$, $(5, 2i)$ and $z_5 = (5,-2i)$. Moreover, by the Theorem on implicit function it appears that $z$ can be chosen as a local coordinate for $z_1$, $z_2$, $z_4$ and $z_5$, and in a same way $w$ can be chosen as a local coordinate for $z_3$.
But now I'm stuck to expand $f(z) = \sqrt{\sqrt{z − 1} − 2}$ with respect to $z$ and $w$:
I believe that $\sqrt{z - 1} = \sqrt{z} - \frac{1}{2}\frac{1}{\sqrt{z}} - \frac{1}{8}\frac{1}{z^3} + \cdots$, but then I'm stuck for $f(z)$.
Also, I'm not sure I understand what is asked when it's said to expand $f(z)$ with respect to $w$...
So if anyone has an idea how to finish this problem, or could tell me if I'm good so far, that vouls be great!
Thanks in advance!