Given an ODE of the form
a regular singular point
is one such that 
or 
diverge as 
, but the limits of 
and 
as exist.
We have for
,
and as
, we have 
and 
, so indeed 
is a regular singular point.
We then look for a series solution about the regular singular point of the form
Substituting into the ODE gives
From this we find the indicial equation to be
Taking
, and in the 
term above we find 
. So we have
Since, all coefficients with an odd index will also vanish.
So the first three terms of the series expansion of this solution are
with
, 
, and 
.
a regular singular point
We have for
and as
We then look for a series solution about the regular singular point
Substituting into the ODE gives
From this we find the indicial equation to be
Taking
Since
So the first three terms of the series expansion of this solution are
with