§13.20 Uniform Asymptotic Approximations for Large $\mu$

When $\mu \to \mathrm{\infty }$ in the sector , with $\kappa (\in ℂ)$ fixed

uniformly for bounded values of $|z|$; also

uniformly for bounded positive values of $x$. For an extension of (13.20.1) to an asymptotic expansion, together with error bounds, see Olver (1997b, Chapter 10, Ex. 3.4).

Let

Then as $\mu \to \mathrm{\infty }$

uniformly with respect to $x\in (0,\mathrm{\infty })$ and $\kappa \in [0,(1-\delta )\mu ]$, where $\delta$ again denotes an arbitrary small positive constant.

Let

with the variable $\zeta$ defined implicitly as follows:

(a) In the case

(b) In the case $\mu =\kappa$

the upper or lower sign being taken according as $x\gtrless 2\mu$.

(In both cases (a) and (b) the $x$-interval $(0,\mathrm{\infty })$ is mapped one-to-one onto the $\zeta$-interval $(-\mathrm{\infty },\mathrm{\infty })$, with $x=0$ and $\mathrm{\infty }$ corresponding to $\zeta =-\mathrm{\infty }$ and $\mathrm{\infty }$, respectively.) Then as $\mu \to \mathrm{\infty }$

uniformly with respect to $x\in (0,\mathrm{\infty })$ and $\kappa \in [-(1-\delta )\mu ,\mu ]$. For the parabolic cylinder function $U$ see §12.2.

These results are proved in Olver (1980b). This reference also supplies error bounds and corresponding approximations when $x$, $\kappa$, and $\mu$ are replaced by $\mathrm{i}x$, $\mathrm{i}\kappa$, and $\mathrm{i}\mu$, respectively.

Again define $\alpha$, $X$, and $\mathrm{\Phi }(\kappa ,\mu ,x)$ by (13.20.6)–(13.20.8), but with $\zeta$ now defined by

when , and by (13.20.10) when $\mu =\kappa$. (As in §13.20(iii) $x=0$ and $\mathrm{\infty }$ correspond to $\zeta =-\mathrm{\infty }$ and $\mathrm{\infty }$, respectively). Then as $\mu \to \mathrm{\infty }$

uniformly with respect to $\zeta \in [0,\mathrm{\infty })$ and $\kappa \in [\mu ,\mu /\delta ]$.

Also,

uniformly with respect to $\zeta \in (-\mathrm{\infty },0]$ and $\kappa \in [\mu ,\mu /\delta ]$.

For the parabolic cylinder functions $U$ and $\overline{U}$ see §12.2, and for the $\mathrm{env}$ functions associated with $U$ and $\overline{U}$ see §14.15(v).

These results are proved in Olver (1980b). Equations (13.20.17) and (13.20.18) are simpler than (6.10) and (6.11) in this reference. Olver (1980b) also supplies error bounds and corresponding approximations when $x$, $\kappa$, and $\mu$ are replaced by $\mathrm{i}x$, $\mathrm{i}\kappa$, and $\mathrm{i}\mu$, respectively.

It should be noted that (13.20.11), (13.20.16), and (13.20.18) differ only in the common error terms. Hence without the error terms the approximation holds for $-(1-\delta )\mu \le \kappa \le \mu /\delta$. Similarly for (13.20.12), (13.20.17), and (13.20.19).

For uniform approximations valid when $\mu$ is large, $x/\mathrm{i}\in (0,\mathrm{\infty })$, and $\kappa /\mathrm{i}\in [0,\mu /\delta ]$, see Olver (1997b, pp. 401–403). These approximations are in terms of Airy functions.

For uniform approximations of $M_{\kappa ,\mathrm{i}\mu }\left(z\right)$ and $W_{\kappa ,\mathrm{i}\mu }\left(z\right)$, $\kappa$ and $\mu$ real, one or both large, see Dunster (2003a).