Another parametrization of (33.2.1) is given by
Again, there is a regular singularity at with indices and , and an irregular singularity of rank 1 at . When the outer turning point is given by
The function is recessive (§2.7(iii)) at , and is defined by
is real and an analytic function of in the interval , and it is also an analytic function of when . This includes , hence can be expanded in a convergent power series in in a neighborhood of (§33.20(ii)).
For nonzero values of and the function is defined by
where is given by (33.14.6) and
(Again, the choice of the ambiguous sign in the last line of (33.14.6) is immaterial.)
is real and an analytic function of each of and in the intervals and , except when or .
The functions and are defined by
An alternative formula for is
the choice of sign in the last line of (33.14.6) again being immaterial.
When and the quantity may be negative, causing and to become imaginary.
The function has the following properties:
where the right-hand side is the Dirac delta (§1.17). When , , is times a polynomial in , and
Note that the functions , , do not form a complete orthonormal system.
With arguments suppressed,