I was reading a book [ Teichmuller Theory and quadratic differential and FarbMargalits' A Primer on MCG ] where they define the natural coordinate of holomorphic quadratic differential on a compact Riemann surface without boundary. They define the natural coordinate by taking the line integral of $ \sqrt \phi $ , where the quadratic differential is $ q = \phi(z) dz^2 $ locally on X. But if $\phi$ has a zero of odd order, how can I define the square root ? In the books they always avoided that case. Any help ?
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.
The following result can be found in the book of Strebel "Quadratic differentials", p. 29. Theorem In the neighborhood of any zero $P_0$ we can introduce a local parameter $\xi$, such that $P_0$ corresponds to $\xi=0$, in terms of which the quadratic differential $q$ has the representation $\phi(\xi) d \xi^2=\big(\frac{n+2}{2} \big)^2 \xi^n d \xi^2$. The integral $\Phi(\xi)$ has then the simple form $\Phi(\xi)=\xi^{\frac{n+2}{2}} + \textrm{const.}$ The proof is straightforward (manipulation of power series), and can be carried out for both $n$ even and $n$ odd. 

