Compute the principal divisors of a hyperlelliptic surface. 
Let $X$ be the hyperelliptic surface defined by $y^2 = x^5-x.$ Note that
  $x$ and $y$ are meromorphic functions on $X.$ Compute the principal divisors
  div($x$) and div($y$).

We have the following definition:

The Divisor of a Meromorphic Function: Principal Divisors. Let $X$
  be a Riemann surface and let $f$ be a meromorphic function on $X$ which is not 
  identically zero. The divisor of $f$, denoted by div($f$), is the divisor defined 
  by the order function: 
  $$div(f)=\sum_{p}Ord_p(f) p$$
  Any divisor of this form is called a principal divisor on $X$. 

Using the following Lemma, I tried a naive approach:

Lemma 1.4. Let $f$ and $g$ be nonzero meromorphic functions on $X.$ Then we have: 



*

*div($fg$)=div($f$)+div($g$).

*div($f/g$)=div($f$)-div($g$).

*div($1/f$)=div(1)-div($f$).        


Observe that div($y^2$)=div($x(x^4-1)$)=div(x)+div($x^4-1$). So, div($x$)=div($y^2$)-div($x^4-1$). Then I am not sure about div($y^2$) and div($x^4-1$). I think they are both zero. 
Also, I have the following at disposal:

Lemma 3.12. Let $\omega$ be a meromorphic 1-form defined in a neighborhood of 
  $p\in X.$ Let $\gamma$ be a small path on $X$ enclosing $p$ and not enclosing any other pole 
  of $\omega.$ Then Res$_p$($\omega$)=$\frac{1}{2\pi i}\int_{\gamma}\omega.$
  Lemma 3.14. Suppose $f$ is a meromorphic function at $p\in X.$ Then $df/f$ is 
  a meromorphic 1-form at $p$, and 
  Res$_p$($df/f$) = ord$_p$($f$). 

Then $Ord_p(y^2)=Res_p(\frac{2}{y})=\frac{1}{2\pi i}\int_{\gamma}\frac{2}{y}.$ But how to solve this integral?
Any suggestions/hints? Thanks.

"... and so we conclude that $div(x)=2P_0-2P_\infty$ and $div(y)=P_0+P_1+P_{-1}+P_i+P_{-i}-5P_\infty$ in the notation explained with exquisite precision above."    
[Excerpt from a manuscript found in a  Klein bottle]
