|
(1) |
This differential equation has an irregular singularity at . It can be solved using the series method
|
(2) |
|
(3) |
Therefore,
|
(4) |
and
|
(5) |
for , 2, .... Since (4) is just a special case of (5),
|
(6) |
for , 1, .... The linearly independent solutions are then
If
, 4, 8, ..., then terminates with the Power , and (normalized so that
the Coefficient of is ) is the regular solution to the equation, known as the Hermite Polynomial. If
, 6, 10, ..., then terminates with the Power , and (normalized so
that the Coefficient of is ) is the regular solution to the equation, known as the Hermite Polynomial.
If , then Hermite's differential equation becomes
|
(9) |
which is of the form
and so has solution
© 1996-9 Eric W. Weisstein
1999-05-25