§14.15 Uniform Asymptotic Approximations

For the interval with fixed $\nu$, real $\mu$, and arbitrary fixed values of the nonnegative integer $J$,

as $\mu \to \mathrm{\infty }$, uniformly with respect to $x$. In other words, the convergent hypergeometric series expansions of ${𝖯}_{\nu }^{-\mu }\left(\pm x\right)$ are also generalized (and uniform) asymptotic expansions as $\mu \to \mathrm{\infty }$, with scale $1/\mathrm{\Gamma }\left(j+1+\mu \right)$, $j=0,1,2,\mathrm{\dots }$; compare §2.1(v).

Provided that $\mu -\nu \notin \mathbb{Z}$ the corresponding expansions for ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mp \mu }\left(x\right)$ can be obtained from the connection formulas (14.9.7), (14.9.9), and (14.9.10).

For the interval the following asymptotic approximations hold when $\mu \to \mathrm{\infty }$, with $\nu$ ($\ge -\frac{1}{2}$) fixed, uniformly with respect to $x$:

where $u$ is given by (14.12.10). Here $I$ and $K$ are the modified Bessel functions (§10.25(ii)).

For asymptotic expansions and explicit error bounds, see Dunster (2003b) and Gil et al. (2000). See also Temme (2015, Chapter 29).

In this and subsequent subsections $\delta$ denotes an arbitrary constant such that .

As $\mu \to \mathrm{\infty }$,

uniformly with respect to $x\in (-1,1)$ and $\nu +\frac{1}{2}\in [0,(1-\delta )\mu ]$, where

and

With the same conditions, the corresponding approximation for ${𝖯}_{\nu }^{-\mu }\left(-x\right)$ is obtained by replacing ${\mathrm{e}}^{-\mu \rho }$ by ${\mathrm{e}}^{\mu \rho }$ on the right-hand side of (14.15.4). Approximations for ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mp \mu }\left(x\right)$ can then be achieved via (14.9.7), (14.9.9), and (14.9.10).

Next,

uniformly with respect to $x\in (1,\mathrm{\infty })$ and $\nu +\frac{1}{2}\in [0,(1-\delta )\mu ]$. Here $\alpha$ is again given by (14.15.5), and $\eta$ is defined implicitly by

The interval is mapped one-to-one to the interval , with the points $x=1$ and $x=\mathrm{\infty }$ corresponding to $\eta =\mathrm{\infty }$ and $\eta =0$, respectively. For asymptotic expansions and explicit error bounds, see Dunster (2003b).

For $\nu \to \mathrm{\infty }$ and fixed $\mu$ ($\ge 0$),

uniformly for $\theta \in (0,\pi -\delta ]$. For the Bessel functions $J$ and $Y$ see §10.2(ii), and for the $env$ functions associated with $J$ and $Y$ see §2.8(iv).

Next,

uniformly for $\xi \in (0,\mathrm{\infty })$.

For asymptotic expansions and explicit error bounds, see Olver (1997b, Chapter 12, §§12, 13) and Jones (2001). For convergent series expansions see Dunster (2004). See also Temme (2015, Chapter 29).

See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials $P_n\left(\cos \theta \right)$ as $n\to \mathrm{\infty }$ with $\theta$ fixed.

As $\nu \to \mathrm{\infty }$,

uniformly with respect to $x\in [0,1)$ and $\mu \in [0,(1-\delta )(\nu +\frac{1}{2})]$. For $\alpha$, $\beta$, and $y$ see below.

Next,

uniformly with respect to $x\in (1,\mathrm{\infty })$ and $\mu \in [0,(1-\delta )(\nu +\frac{1}{2})]$. In (14.15.15)–(14.15.18)

and the variable $y$ is defined implicitly by

and

where the inverse trigonometric functions take their principal values (§4.23(ii)). The points $x={\left(1-{\alpha }^2\right)}^{1/2}$, $x=1$, and $x=\mathrm{\infty }$ are mapped to $y={\alpha }^2$, $y=0$, and $y=-\mathrm{\infty }$, respectively. The interval is mapped one-to-one to the interval , where $y=y_0$ is the (positive) solution of (14.15.21) when $x=0$.

For asymptotic expansions and explicit error bounds, see Boyd and Dunster (1986).

Here we introduce the envelopes of the parabolic cylinder functions $U(-c,x)$, $\overline{U}(-c,x)$, which are defined in §12.2. For $U(-c,x)$ or $\overline{U}(-c,x)$, with $c$ and $x$ nonnegative,

where $x=X_c$ denotes the largest positive root of the equation $U(-c,x)=\overline{U}(-c,x)$.

As $\nu \to \mathrm{\infty }$,

uniformly with respect to $x\in [0,1)$ and $\mu \in [\delta (\nu +\frac{1}{2}),\nu +\frac{1}{2}]$. Here

and the variable $\zeta$ is defined implicitly by

and

when $a>0$, and

when $a=0$. The inverse hyperbolic and trigonometric functions take their principal values (§§4.23(ii), 4.37(ii)).

When $a>0$ the interval is mapped one-to-one to the interval , with the points $x=-a$, $x=a$, and $x=1$ corresponding to $\zeta =-\alpha$, $\zeta =\alpha$, and $\zeta =\mathrm{\infty }$, respectively. When $a=0$ the interval is mapped one-to-one to the interval , with the points $x=-1$, $0$, and $1$ corresponding to $\zeta =-\mathrm{\infty }$, $0$, and $\mathrm{\infty }$, respectively.

Next, as $\nu \to \mathrm{\infty }$,

uniformly with respect to $x\in (-1,1)$ and $\mu \in [\nu +\frac{1}{2},(1/\delta )(\nu +\frac{1}{2})]$. Here $\zeta$ is defined implicitly by

when $a>0$, which maps the interval one-to-one to the interval : the points $x=-1$ and $x=1$ correspond to $\zeta =-\mathrm{\infty }$ and $\zeta =\mathrm{\infty }$, respectively. When $a=0$ (14.15.29) again applies. (The inverse hyperbolic functions again take their principal values.)

Since (14.15.30) holds for negative $x$, corresponding approximations for ${𝖰}_{\nu }^{\mp \mu }\left(x\right)$, uniformly valid in the interval , can be obtained from (14.9.9) and (14.9.10).

For error bounds and other extensions see Olver (1975b).