§33.14 Definitions and Basic Properties

Another parametrization of (33.2.1) is given by

where

Again, there is a regular singularity at $r=0$ with indices $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $r=\mathrm{\infty }$. When $ϵ>0$ the outer turning point is given by

compare (33.2.2).

The function $f(ϵ,\mathrm{\ell };r)$ is recessive (§2.7(iii)) at $r=0$, and is defined by

or equivalently

where $M_{\kappa ,\mu }\left(z\right)$ and $M(a,b,z)$ are defined in §§13.14(i) and 13.2(i), and

The choice of sign in the last line of (33.14.6) is immaterial: the same function $f(ϵ,\mathrm{\ell };r)$ is obtained. This is a consequence of Kummer’s transformation (§13.2(vii)).

$f(ϵ,\mathrm{\ell };r)$ is real and an analytic function of $r$ in the interval , and it is also an analytic function of $ϵ$ when . This includes $ϵ=0$, hence $f(ϵ,\mathrm{\ell };r)$ can be expanded in a convergent power series in $ϵ$ in a neighborhood of $ϵ=0$ (§33.20(ii)).

For nonzero values of $ϵ$ and $r$ the function $h(ϵ,\mathrm{\ell };r)$ is defined by

where $\kappa$ is given by (33.14.6) and

(Again, the choice of the ambiguous sign in the last line of (33.14.6) is immaterial.)

$h(ϵ,\mathrm{\ell };r)$ is real and an analytic function of each of $r$ and $ϵ$ in the intervals and , except when $r=0$ or $ϵ=0$.

The functions $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ are defined by

where

and

An alternative formula for $A(ϵ,\mathrm{\ell })$ is

the choice of sign in the last line of (33.14.6) again being immaterial.

When and $\mathrm{\ell }>{(-ϵ)}^{-1/2}$ the quantity $A(ϵ,\mathrm{\ell })$ may be negative, causing $s(ϵ,\mathrm{\ell };r)$ and $c(ϵ,\mathrm{\ell };r)$ to become imaginary.

The function $s(ϵ,\mathrm{\ell };r)$ has the following properties:

where the right-hand side is the Dirac delta (§1.17). When $ϵ=-1/n^2$, $n=\mathrm{\ell }+1,\mathrm{\ell }+2,\mathrm{\dots }$, $s(ϵ,\mathrm{\ell };r)$ is $\exp \left(-r/n\right)$ times a polynomial in $r/n$, and

satisfies

Note that the functions ${\varphi }_{n,\mathrm{\ell }}$, $n=\mathrm{\ell },\mathrm{\ell }+1,\mathrm{\dots }$, do not form a complete orthonormal system.

With arguments $ϵ,\mathrm{\ell },r$ suppressed,