Let $X$ be a hyperelliptic Riemann surface defined by the zero set of the function $f(x,y) = y^2 - x^5 +x$. I would like to compute the principal divisors of the meromorphic function $x$ and $y$.
In general, if $g$ is a meromorphic function on some Riemann surface $\Sigma$, then $$ \operatorname{div}(g) = \sum_{p \in \Sigma} \operatorname{ord}_p(g) \cdot p. $$ If $\phi: U \to \phi(U)$ is a chart of $\Sigma$ around $p$, then $\operatorname{ord}_p(g) = \min\{n: c_n \neq 0\}$, where $c_n$ is the $n$th coefficient of the Laurent expansion at $p$ of the complex-analytic function $g \circ \phi^{-1}: \phi(U) \to \mathbb{C}$.
Since $h(x) = x^5-x$ has degree $5 = 2g+1$, the genus of $X$ is 2. Define the function $k(z) = z^{6}h(1/z)$, which is a polynomial in $z$. Then, $X = X_0 \cup_\beta X_1$, where $$X_0 = \{(x,y) \in \mathbb{C}^2: y^2 = h(x)\}$$ $$X_1 = \{(z,w) \in \mathbb{C}^2: w^2 = k(z)\},$$ and $\beta: U_0 = \{x \neq 0\} \xrightarrow{\sim} V_0 = \{z \neq 0\}$ given by $\beta(x,y) = (x^{-1}, yx^{-3})$ is the attaching map.
Here begins my reasoning.
a) Computing the principal divisor of $x$.
For a zero $p = (0,y)$ of $x$, necessarily, $y = 0$, so we obtain $\operatorname{ord}_{(0,0)}(x) = \operatorname{ord}_{(0,0)}(y^2) = 2$.
To determine the order at $\infty$, we substitute $z = x^{-1}$ and $w = yx^{-3}$ on the open set $U_0$, where $x \neq 0$. Then, $w^2 = z(z^4-1)$ by what has been said above, and $x = \infty$ if and only if $z = 0$ if and only $w^2 = 0$. In other words, $\operatorname{ord}_{\infty}(x) = 2$. We conclude:
$$ \operatorname{div}(x) = 2 \cdot 0 - 2 \cdot \infty. $$
b) Computing the principal divisor of $y$.
Clearly, $y = 0$ implies $x(x^4-1) = 0$, hence, $y$ (defined as a meromorphic function on $X$) vanishes at the points $(0,0)$ and $e^{2\pi ik/4}$ for $k =0,1,2,3$. Each of these points must have order 1 since $y$ has degree $5$.
I am a bit stuck with the poles of $y$. I would like to perform a coordinate change as previously, that is $w = 1/y$. But then what? Since $\deg(\operatorname{div}(y)) = 0$, I already know that $y$ has five poles counted with multiplicity. A little hint to resolve this issue is much appreciated.