Bessel’s equation of order is given by
(10.6) |
Clearly is a regular singular point, so we can solve by Frobenius’ method. The indicial equation is obtained from the lowest power after the substitution , and is
(10.7) |
So a generalised series solution gives two independent solutions if . Now let us solve the problem and explicitly substitute the power series,
(10.8) |
From Bessel’s equation we find
(10.9) |
which leads to
(10.10) |
or
(10.11) |
If we take , we have
(10.12) |
This can be solved by iteration,
If we choose 1 we find the Bessel function of order
(10.14) |
There is another second independent solution (which should have a logarithm in it) with goes to infinity at .
The general solution of Bessel’s equation of order is a linear combination of and ,
(10.15) |