Defining the natural co-ordinate for a holomorphic quadratic differential near a zero of odd order

Question

I was reading a book [ Teichmuller Theory and quadratic differential and Farb-Margalits' A Primer on MCG ] where they define the natural co-ordinate of holomorphic quadratic differential on a compact Riemann surface without boundary. They define the natural co-ordinate 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 ?

Answer 1

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.