is the bessel equation.
with 
. Lots of algebra gives 
and
a recurrence relation, 
. Also 
. This means 
where 
is the gamma
function. Defining 
and 
, you get: 
. To get the second linearly independent solution
when , you take the derivative with respect of 
of the Bessel equation,
and get the solution: 
for . To get a solution always, redefine
for any . Typically, use, 
.
as x- 
0
. these are singular.
for x 
for x 
. This lets you get an expansion of 
.
where 
is the nth root of 
if 
= 0 of 
0 of 
with 