The function is recessive (§2.7(iii)) at , and is defined by
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
The functions are defined by
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 .
With arguments suppressed,