§33.12 Asymptotic Expansions for Large $\eta$

When $\mathrm{\ell }=0$ and $\eta >0$, the outer turning point is given by ${\rho }_{\mathrm{tp}}(\eta ,0)=2\eta$; compare (33.2.2). Define

Then as $\eta \to \mathrm{\infty }$,

uniformly for bounded values of $\left|(\rho -2\eta )/{\eta }^{1/3}\right|$. Here $\mathrm{Ai}$ and $\mathrm{Bi}$ are the Airy functions (§9.2), and

In particular,

where $\omega ={(\frac{2}{3}\eta )}^{1/3}$.

For derivations and additional terms in the expansions in this subsection see Abramowitz and Rabinowitz (1954) and Fröberg (1955). For asymptotic expansions of $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ when $\eta \to \pm \mathrm{\infty }$ see Temme (2015, Chapter 31).

With the substitution $\rho =2\eta z$, Equation (33.2.1) becomes

Then, by application of the results given in §§2.8(iii) and 2.8(iv), two sets of asymptotic expansions can be constructed for $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ when $\eta \to \mathrm{\infty }$. See Temme (2015, §31.7).

The first set is in terms of Airy functions and the expansions are uniform for fixed $\mathrm{\ell }$ and , where $\delta$ is an arbitrary small positive constant. They would include the results of §33.12(i) as a special case.

The second set is in terms of Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$, and they are uniform for fixed $\mathrm{\ell }$ and $0\le z\le 1-\delta$, where $\delta$ again denotes an arbitrary small positive constant.

Compare also §33.20(iv).