§10.41 Asymptotic Expansions for Large Order

If $\nu \to \mathrm{\infty }$ through positive real values with $z(\ne 0)$ fixed, then

As $\nu \to \mathrm{\infty }$ through positive real values,

uniformly for . Here

where the branches assume their principal values. Also, $U_k(p)$ and $V_k(p)$ are polynomials in $p$ of degree $3k$, given by $U_0(p)=V_0(p)=1$, and

For $k=1,2,3$,

For $U_4(p)$, $U_5(p)$, $U_6(p)$, see Bickley et al. (1952, p. xxxv).

For numerical tables of $\eta =\eta (z)$ and the coefficients $U_k(p)$, $V_k(p)$, see Olver (1962, pp. 43–51).

The expansions (10.41.3)–(10.41.6) also hold uniformly in the sector $|\mathrm{ph}z|\le \frac{1}{2}\pi -\delta$ , with the branches of the fractional powers in (10.41.3)–(10.41.8) extended by continuity from the positive real $z$-axis.

Figures 10.41.1 and 10.41.2 show corresponding points of the mapping of the $z$-plane and the $\eta$-plane. The curve $E_{1}B{E}_2$ in the $z$-plane is the upper boundary of the domain $𝐊$ depicted in Figure 10.20.3 and rotated through an angle $-\frac{1}{2}\pi$. Thus $B$ is the point $z=c$, where $c$ is given by (10.20.18).

For derivations of the results in this subsection, and also error bounds, see Olver (1997b, pp. 374–378). For extensions of the regions of validity in the $z$-plane and extensions to complex values of $\nu$ see Olver (1997b, pp. 378–382).

For expansions in inverse factorial series see Dunster et al. (1993).

The series (10.41.3)–(10.41.6) can also be regarded as generalized asymptotic expansions for large $|z|$. Thus as $z\to \mathrm{\infty }$ with $\mathrm{\ell }$ $(\ge 1)$ and $\nu$ $(>0)$ both fixed,

Similarly for (10.41.5) and (10.41.6).

In the case of (10.41.13) with positive real values of $z$ the result is a consequence of the error bounds given in Olver (1997b, pp. 377–378). Then by expanding the quantities $\eta$, ${(1+z^2)}^{-\frac{1}{4}}$, and $U_k(p)$, $k=0,1,\mathrm{\dots },\mathrm{\ell }-1$, and rearranging, we arrive at an expansion of the right-hand side of (10.41.13) in powers of $z^{-1}$. Moreover, because of the uniqueness property of asymptotic expansions (§2.1(iii)) this expansion must agree with (10.40.2), with $z$ replaced by $\nu z$, up to and including the term in $z^{-(\mathrm{\ell }-1)}$. It also enjoys the same sector of validity.

To establish (10.41.12) we substitute into (10.34.3), with $m=0$ and $z$ replaced by $\nu z$, by means of (10.41.13) observing that when $|z|$ is large the effect of replacing $z$ by $z{\mathrm{e}}^{\pm \pi \mathrm{i}}$ is to replace $\eta$, ${(1+z^2)}^{\frac{1}{4}}$, and $p$ by $-\eta$, $\pm \mathrm{i}{(1+z^2)}^{\frac{1}{4}}$, and $-p$, respectively.

Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. We first prove that for the expansions (10.20.6) for the Hankel functions $H_{\nu }^{(1)}\left(\nu z\right)$ and $H_{\nu }^{(2)}\left(\nu z\right)$ the $z$-asymptotic property applies when $z\to \pm \mathrm{i}\mathrm{\infty }$, respectively. This is a consequence of the error bounds associated with these expansions. We then extend the validity of this property from $z\to \pm \mathrm{i}\mathrm{\infty }$ to $z\to \mathrm{\infty }$ in the sector $-\pi +\delta \le \mathrm{ph}z\le 2\pi -\delta$ in the case of $H_{\nu }^{(1)}\left(\nu z\right)$, and to $z\to \mathrm{\infty }$ in the sector $-2\pi +\delta \le \mathrm{ph}z\le \pi -\delta$ in the case of $H_{\nu }^{(2)}\left(\nu z\right)$. This is done by re-expansion with the aid of (10.20.10), (10.20.11), and §10.41(ii), followed by comparison with (10.17.5) and (10.17.6), with $z$ replaced by $\nu z$. Lastly, we substitute into (10.4.4), again with $z$ replaced by $\nu z$. The final results are:

as $z\to \mathrm{\infty }$ in $|\mathrm{ph}z|\le \pi -\delta$, or equivalently as $\zeta \to \mathrm{\infty }$ in $|\mathrm{ph}\left(-\zeta \right)|\le \frac{2}{3}\pi -\delta$, for fixed $\mathrm{\ell }$ $(\ge 0)$ and fixed $\nu$ $(>0)$.

It needs to be noted that the results (10.41.14) and (10.41.15) do not apply when $z\to 0+$ or equivalently $\zeta \to +\mathrm{\infty }$. This is because $A_k(\zeta )$ and ${\zeta }^{-\frac{1}{2}}{B}_k(\zeta ),k=0,1,\mathrm{\dots }$, do not form an asymptotic scale (§2.1(v)) as $\zeta \to +\mathrm{\infty }$; see Olver (1997b, pp. 422–425).