# §33.2 Definitions and Basic Properties

## §33.2(i) Coulomb Wave Equation

This differential equation has a regular singularity at with indices and , and an irregular singularity of rank 1 at (§§2.7(i), 2.7(ii)). There are two turning points, that is, points at which 2.8(i)). The outer one is given by

33.2.2

## §33.2(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

33.2.5

The choice of ambiguous signs in (33.2.3) and (33.2.4) is immaterial, provided that either all upper signs are taken, or all lower signs are taken. This is a consequence of Kummer’s transformation (§13.2(vii)).

is a real and analytic function of on the open interval , and also an analytic function of when .

The normalizing constant is always positive, and has the alternative form

## §33.2(iii) Irregular Solutions

The functions are defined by

or equivalently

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

33.2.9

and

33.2.10

the branch of the phase in (33.2.10) being zero when and continuous elsewhere. is the Coulomb phase shift.

and are complex conjugates, and their real and imaginary parts are given by

As in the case of , the solutions and are analytic functions of when . Also, are analytic functions of when .