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I'm trying to find the germs $[z,f]$ for the complete holomorphic function $\sqrt{1+\sqrt{z}}$ . The question indicates that I should find 2 germs for z = 1, but I seem to be able to find 3! Where have I gone wrong?

I start by defining $U_k={\frac{k\pi}{2}<\theta<\pi+\frac{k\pi}{2}}$, then ${(U_k, f_k)}$ for $0\leq k \leq 15$ is an explicit formulation of the complete holomorphic function, where $f_k = f$.

Now let $B$ be a small open ball around $z=1$. Clearly

$f_0(1) = \sqrt{2}$, $f_4(1) = 0$, $f_8(1) = -\sqrt{2}$, $f_{12}(1)=0$

and in particular we have $f_0 \ne f_4 \ne f_8$ on $B$ so I have at least 3 different germs! But in fact there are exactly 3 as it's easy to see that $f_4 =f_{12}$ on $B$.

Presumably my reasoning about $f_0$ and $f_8$ is somehow wrong - could someone help me out? Many thanks in advance!

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When you say $f_k = f$, I think you mean something along the lines of "$f_k = $ the formula for $f$ where the radicals are interpreted according to some natural branch cut of the square root function associated with the set $U_k$," am I reading you right? Can you make the branch cuts explicit? –  user29743 Apr 25 '12 at 20:05
    
That is indeed what I mean. Hmm yes maybe making them explicit is where I fall down. I assumed I could simply write $z=e^{i\theta}$ for $\theta$ in an appropriate range for the relevant $U_k$. Certainly then $\sqrt{z}$ is well defined. Then can I just write $1=e^{i\phi}$ for $\phi$ in the appropriate range also? Or is this not well-defined? If that is where I'm falling down, do you have an alternative suggestion for how to solve the problem! Thanks! –  Edward Hughes Apr 25 '12 at 20:19
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