6.2.1.2 step 2

Using quantities calculated in step \(1\), the algorithm now searches for a non-negative integer \(d\) using

\begin{align*} d&=\alpha _{\infty }^{\pm }-\sum _{c\in \Gamma }\alpha _{c}^{\pm } \end{align*}

If non-negative \(d\) is found, a candidate \(\omega _{d}\) is calculated using

\begin{align*} \omega _{d}&=\sum _{c\in \Gamma }\left ( (\pm ) \left [ \sqrt {r}\right ] _{c}+\frac {\alpha _{c}^{\pm }}{x-c}\right ) + (\pm ) \left [\sqrt {r}\right ]_{\infty } \end{align*}

If no non-negative integer \(d\) could be found, then no Liouvillian solution exists using this case. Case two or three are tried next if these are available.