There are solutions of (32.2.1) such that
and and are constants.
There are also solutions of (32.2.1) such that
Next, for given initial conditions and , with real, has at least one pole on the real axis. There are two special values of , and , with the properties , , and such that:
If , then for , where is the first pole on the negative real axis.
If , then oscillates about, and is asymptotic to, as .
If , then changes sign once, from positive to negative, as passes from to .
Consider the special case of with :
with boundary condition
Any nontrivial real solution of (32.11.4) that satisfies (32.11.5) is asymptotic to , for some nonzero real , where denotes the Airy function (§9.2). Conversely, for any nonzero real , there is a unique solution of (32.11.4) that is asymptotic to as .
If , then exists for all sufficiently large as , and
and , are real constants. Connection formulas for and are given by
where is the gamma function (§5.2(i)), and the branch of the function is immaterial.
If , then
If , then has a pole at a finite point , dependent on , and
Replacement of by in (32.11.4) gives
Any nontrivial real solution of (32.11.12) satisfies
with and arbitrary real constants.
In the case when
with , we have
where is a nonzero real constant. The connection formulas for are
In the generic case
where , , and are real constants, and
The connection formulas for , , and are
For , with and ,
where is an arbitrary constant such that , and
Consider with and , that is,
and with boundary condition
Any nontrivial solution of (32.11.29) that satisfies (32.11.30) is asymptotic to as , where is a constant. Conversely, for any there is a unique solution of (32.11.29) that is asymptotic to as . Here denotes the parabolic cylinder function (§12.2).
Now suppose . If , where
then has no poles on the real axis. Furthermore, if , then
Alternatively, if is not zero or a positive integer, then
and and are real constants. Connection formulas for and are given by
and the branch of the function is immaterial.
Next if , then
and has no poles on the real axis.
Lastly if , then has a simple pole on the real axis, whose location is dependent on .
See also Wong and Zhang (2009a).