# §9.7 Asymptotic Expansions

## §9.7(i) Notation

Here $\delta$ denotes an arbitrary small positive constant and

 9.7.1 $\zeta=\tfrac{2}{3}z^{\ifrac{3}{2}}.$ ⓘ Defines: $\zeta(z)$: change of variable (locally) Symbols: $z$: complex variable Permalink: http://dlmf.nist.gov/9.7.E1 Encodings: TeX, pMML, png See also: Annotations for 9.7(i), 9.7 and 9

Also $u_{0}=v_{0}=1$ and for $k=1,2,\ldots,$

 9.7.2 $\displaystyle u_{k}$ $\displaystyle=\frac{(2k+1)(2k+3)(2k+5)\cdots(6k-1)}{216^{k}k!}=\frac{(6k-5)(6k% -3)(6k-1)}{(2k-1)216k}u_{k-1},$ $\displaystyle v_{k}$ $\displaystyle=\frac{6k+1}{1-6k}u_{k}.$ ⓘ Symbols: $!$: factorial (as in $n!$), $k$: nonnegative integer, $u_{s}$: expansion coefficient and $v_{s}$: expansion coefficient Source: Olver (1997b, p. 392) Referenced by: Other Changes Permalink: http://dlmf.nist.gov/9.7.E2 Encodings: TeX, TeX, pMML, pMML, png, png Addition (effective with 1.0.16): The recurrence relation $u_{k}=\frac{(6k-5)(6k-3)(6k-1)}{(2k-1)216k}u_{k-1}$ was added to this equation. Reported 2017-04-06 by James McTavish See also: Annotations for 9.7(i), 9.7 and 9

Lastly,

 9.7.3 $\chi(n)=\pi^{\ifrac{1}{2}}\Gamma\left(\tfrac{1}{2}n+1\right)/\Gamma\left(% \tfrac{1}{2}n+\tfrac{1}{2}\right).$ ⓘ Defines: $\chi(n)$: function (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function and $\pi$: the ratio of the circumference of a circle to its diameter Source: Olver (1997b, (13.02), p. 225) Permalink: http://dlmf.nist.gov/9.7.E3 Encodings: TeX, pMML, png See also: Annotations for 9.7(i), 9.7 and 9

Numerical values of this function are given in Table 9.7.1 for $n=1(1)20$ to 2D. For large $n$,

 9.7.4 $\chi(n)\sim(\tfrac{1}{2}\pi n)^{\ifrac{1}{2}}.$ ⓘ Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $\chi(n)$: function Source: Olver (1997b, p. 225) Permalink: http://dlmf.nist.gov/9.7.E4 Encodings: TeX, pMML, png See also: Annotations for 9.7(i), 9.7 and 9

## §9.7(ii) Poincaré-Type Expansions

As $z\to\infty$ the following asymptotic expansions are valid uniformly in the stated sectors.

 9.7.5 $\displaystyle\mathrm{Ai}\left(z\right)$ $\displaystyle\sim\frac{e^{-\zeta}}{2\sqrt{\pi}z^{1/4}}\sum_{k=0}^{\infty}(-1)^% {k}\frac{u_{k}}{\zeta^{k}},$ $|\operatorname{ph}z|\leq\pi-\delta$, 9.7.6 $\displaystyle\mathrm{Ai}'\left(z\right)$ $\displaystyle\sim-\frac{z^{1/4}e^{-\zeta}}{2\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)% ^{k}\frac{v_{k}}{\zeta^{k}},$ $|\operatorname{ph}z|\leq\pi-\delta$,
 9.7.7 $\displaystyle\mathrm{Bi}\left(z\right)$ $\displaystyle\sim\frac{e^{\zeta}}{\sqrt{\pi}z^{1/4}}\sum_{k=0}^{\infty}\frac{u% _{k}}{\zeta^{k}},$ $|\operatorname{ph}z|\leq\tfrac{1}{3}\pi-\delta$, 9.7.8 $\displaystyle\mathrm{Bi}'\left(z\right)$ $\displaystyle\sim\frac{z^{1/4}e^{\zeta}}{\sqrt{\pi}}\sum_{k=0}^{\infty}\frac{v% _{k}}{\zeta^{k}},$ $|\operatorname{ph}z|\leq\tfrac{1}{3}\pi-\delta$.
 9.7.9 $\displaystyle\mathrm{Ai}\left(-z\right)$ $\displaystyle\sim\frac{1}{\sqrt{\pi}z^{1/4}}\left(\cos\left(\zeta-\tfrac{1}{4}% \pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k}}{\zeta^{2k}}+\sin\left(\zeta% -\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{2k+1}% }\right),$ $|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta$, 9.7.10 $\displaystyle\mathrm{Ai}'\left(-z\right)$ $\displaystyle\sim\frac{z^{1/4}}{\sqrt{\pi}}\left(\sin\left(\zeta-\tfrac{1}{4}% \pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{2k}}{\zeta^{2k}}-\cos\left(\zeta% -\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{2k+1}}{\zeta^{2k+1}% }\right),$ $|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta$,
 9.7.11 $\displaystyle\mathrm{Bi}\left(-z\right)$ $\displaystyle\sim\frac{1}{\sqrt{\pi}z^{1/4}}\left(-\sin\left(\zeta-\tfrac{1}{4% }\pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k}}{\zeta^{2k}}+\cos\left(% \zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{% 2k+1}}\right),$ $|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta$, 9.7.12 $\displaystyle\mathrm{Bi}'\left(-z\right)$ $\displaystyle\sim\frac{z^{1/4}}{\sqrt{\pi}}\left(\cos\left(\zeta-\tfrac{1}{4}% \pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{2k}}{\zeta^{2k}}+\sin\left(\zeta% -\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{2k+1}}{\zeta^{2k+1}% }\right),$ $|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta$.
 9.7.13 $\mathrm{Bi}\left(ze^{\pm\pi i/3}\right)\mathrel{\sim}\sqrt{\frac{2}{\pi}}\frac% {e^{\pm\pi i/6}}{z^{1/4}}\*\left(\cos\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2% }\mathrm{i}\ln 2\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k}}{\zeta^{2k}}+% \sin\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}\mathrm{i}\ln 2\right)\sum_{k=0}% ^{\infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{2k+1}}\right),$ $|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta$,
 9.7.14 $\mathrm{Bi}'\left(ze^{\pm\pi i/3}\right)\mathrel{\sim}\sqrt{\frac{2}{\pi}}e^{% \mp\pi i/6}z^{1/4}\*\left(-\sin\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}% \mathrm{i}\ln 2\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{2k}}{\zeta^{2k}}+% \cos\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}\mathrm{i}\ln 2\right)\sum_{k=0}% ^{\infty}(-1)^{k}\frac{v_{2k+1}}{\zeta^{2k+1}}\right),$ $|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta$.

## §9.7(iii) Error Bounds for Real Variables

In (9.7.5) and (9.7.6) the $n$th error term, that is, the error on truncating the expansion at $n$ terms, is bounded in magnitude by the first neglected term and has the same sign, provided that the following term is of opposite sign, that is, if $n\geq 0$ for (9.7.5) and $n\geq 1$ for (9.7.6).

In (9.7.7) and (9.7.8) the $n$th error term is bounded in magnitude by the first neglected term multiplied by $2\chi(n)\exp\left(\sigma\pi/(72\zeta)\right)$ where $\sigma=5$ for (9.7.7) and $\sigma=7$ for (9.7.8), provided that $n\geq 1$ in both cases.

In (9.7.9)–(9.7.12) the $n$th error term in each infinite series is bounded in magnitude by the first neglected term and has the same sign, provided that the following term in the series is of opposite sign.

As special cases, when $0

 9.7.15 $\displaystyle\mathrm{Ai}\left(x\right)$ $\displaystyle\leq\frac{e^{-\xi}}{2\sqrt{\pi}x^{1/4}}$, $\displaystyle|\mathrm{Ai}'\left(x\right)|$ $\displaystyle\leq\frac{x^{1/4}e^{-\xi}}{2\sqrt{\pi}}\left(1+\frac{7}{72\xi}\right)$, ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $x$: real variable and $xi$: change of variable Source: Olver (1997b, p. 394) Permalink: http://dlmf.nist.gov/9.7.E15 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.7(iii), 9.7 and 9
 9.7.16 $\displaystyle\mathrm{Bi}\left(x\right)$ $\displaystyle\leq\frac{e^{\xi}}{\sqrt{\pi}x^{1/4}}\left(1+\frac{5\pi}{72\xi}% \exp\left(\frac{5\pi}{72\xi}\right)\right),$ $\displaystyle\mathrm{Bi}'\left(x\right)$ $\displaystyle\leq\frac{x^{1/4}e^{\xi}}{\sqrt{\pi}}\left(1+\frac{7\pi}{72\xi}% \exp\left(\frac{7\pi}{72\xi}\right)\right),$ ⓘ Symbols: $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp\NVar{z}$: exponential function, $\mathrm{e}$: base of exponential function, $x$: real variable and $xi$: change of variable Source: Derivable from Olver (1997b, p. 394). Permalink: http://dlmf.nist.gov/9.7.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.7(iii), 9.7 and 9

where $\xi=\tfrac{2}{3}x^{3/2}$.

## §9.7(iv) Error Bounds for Complex Variables

When $n\geq 1$ the $n$th error term in (9.7.5) and (9.7.6) is bounded in magnitude by the first neglected term multiplied by

 9.7.17 $\begin{cases}2\exp\left(\dfrac{\sigma}{36|\zeta|}\right)&|\operatorname{ph}z|% \leq\tfrac{1}{3}\pi,\\ 2\chi(n)\exp\left(\dfrac{\sigma\pi}{72|\zeta|}\right)&\tfrac{1}{3}\pi\leq|% \operatorname{ph}z|\leq\tfrac{2}{3}\pi,\\ \dfrac{4\chi(n)}{|\cos\left(\operatorname{ph}\zeta\right)|^{n}}\exp\left(% \dfrac{\sigma\pi}{36|\Re\zeta|}\right)&\tfrac{2}{3}\pi\leq|\operatorname{ph}z|% <\pi.\end{cases}$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\exp\NVar{z}$: exponential function, $\operatorname{ph}$: phase, $\Re$: real part, $z$: complex variable, $\zeta(z)$: change of variable, $\chi(n)$: function, $n$: index and $\sigma$: index Source: Derivable from (9.6.1)–(9.6.5) and Olver (1997b, pp. 266–267). Referenced by: Equation (9.7.17) Permalink: http://dlmf.nist.gov/9.7.E17 Encodings: TeX, pMML, png Errata (effective with 1.0.11): Originally the constraint condition $\frac{2}{3}\pi\leq|\operatorname{ph}z|<\pi$ was written incorrectly as $\frac{2}{3}\pi\leq|\operatorname{ph}z|\leq\pi$. Also, the equation was reformatted to display the constraints in the equation instead of in the text. Reported 2014-11-05 by Gergő Nemes See also: Annotations for 9.7(iv), 9.7 and 9

Here $\sigma=5$ for (9.7.5) and $\sigma=7$ for (9.7.6).

Corresponding bounds for the errors in (9.7.7) to (9.7.14) may be obtained by use of these results and those of §9.2(v) and their differentiated forms.

For other error bounds see Boyd (1993).

## §9.7(v) Exponentially-Improved Expansions

In (9.7.5) and (9.7.6) let

 9.7.18 $\displaystyle\mathrm{Ai}\left(z\right)$ $\displaystyle=\frac{e^{-\zeta}}{2\sqrt{\pi}z^{1/4}}\left(\sum_{k=0}^{n-1}(-1)^% {k}\frac{u_{k}}{\zeta^{k}}+R_{n}(z)\right),$ 9.7.19 $\displaystyle\mathrm{Ai}'\left(z\right)$ $\displaystyle=-\frac{z^{1/4}e^{-\zeta}}{2\sqrt{\pi}}\left(\sum_{k=0}^{n-1}(-1)% ^{k}\frac{v_{k}}{\zeta^{k}}+S_{n}(z)\right),$

with $n=\left\lfloor 2|\zeta|\right\rfloor$. Then

 9.7.20 $\displaystyle R_{n}(z)$ $\displaystyle=(-1)^{n}\sum_{k=0}^{m-1}(-1)^{k}u_{k}\frac{G_{n-k}\left(2\zeta% \right)}{\zeta^{k}}+R_{m,n}(z),$ ⓘ Defines: $R_{n}$: remainder function (locally) Symbols: $G_{\NVar{p}}\left(\NVar{z}\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: change of variable, $m$: index, $n$: index and $u_{s}$: expansion coefficient Source: Use the methods of Olver (1991b, 1993a). Permalink: http://dlmf.nist.gov/9.7.E20 Encodings: TeX, pMML, png See also: Annotations for 9.7(v), 9.7 and 9 9.7.21 $\displaystyle S_{n}(z)$ $\displaystyle=(-1)^{n-1}\sum_{k=0}^{m-1}(-1)^{k}v_{k}\frac{G_{n-k}\left(2\zeta% \right)}{\zeta^{k}}+S_{m,n}(z),$ ⓘ Defines: $S_{n}$: remainder function (locally) Symbols: $G_{\NVar{p}}\left(\NVar{z}\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: change of variable, $m$: index, $n$: index and $v_{s}$: expansion coefficient Source: Use the methods of Olver (1991b, 1993a). Permalink: http://dlmf.nist.gov/9.7.E21 Encodings: TeX, pMML, png See also: Annotations for 9.7(v), 9.7 and 9

where

 9.7.22 $G_{p}\left(z\right)=\frac{e^{z}}{2\pi}\Gamma\left(p\right)\Gamma\left(1-p,z% \right).$ ⓘ Defines: $G_{\NVar{p}}\left(\NVar{z}\right)$: rescaled terminant function Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function, $\Gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function and $z$: complex variable Source: Use the methods of Olver (1991b, 1993a). Permalink: http://dlmf.nist.gov/9.7.E22 Encodings: TeX, pMML, png See also: Annotations for 9.7(v), 9.7 and 9

(For the notation see §8.2(i).) And as $z\rightarrow\infty$ with $m$ fixed

 9.7.23 $R_{m,n}(z),S_{m,n}(z)=O\left(e^{-2|\zeta|}\zeta^{-m}\right),$ $|\operatorname{ph}z|\leq\tfrac{2}{3}\pi$.

For re-expansions of the remainder terms in (9.7.7)–(9.7.14) combine the results of this section with those of §9.2(v) and their differentiated forms, as in §9.7(iv).

For higher re-expansions of the remainder terms see Olde Daalhuis (1995, 1996), and Olde Daalhuis and Olver (1995a).