# §33.14 Definitions and Basic Properties

## §33.14(i) Coulomb Wave Equation

Another parametrization of (33.2.1) is given by

where

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

33.14.3

compare (33.2.2).

## §33.14(ii) Regular Solution

The function is recessive (§2.7(iii)) at , and is defined by

or equivalently

where and are defined in §§13.14(i) and 13.2(i), and

33.14.6

The choice of sign in the last line of (33.14.6) is immaterial: the same function is obtained. This is a consequence of Kummer’s transformation (§13.2(vii)).

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)).

## §33.14(iii) Irregular Solution

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 .

## §33.14(iv) Solutions and

The functions and are defined by

33.14.9

provided that when , where

and

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

33.14.14

satisfies