# §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: $\zeta$: 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 Ch.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: Erratum (V1.0.16) for Equation (9.7.2), Erratum (V1.0.16) for Equation (9.7.2) 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. Suggested 2017-04-06 by James McTavish See also: Annotations for §9.7(i), §9.7 and Ch.9

Lastly, for $x>0$ we define

 9.7.3 $\chi(x)\equiv{\pi}^{\ifrac{1}{2}}\Gamma\left(\tfrac{1}{2}x+1\right)/\Gamma% \left(\tfrac{1}{2}x+\tfrac{1}{2}\right).$ ⓘ Defines: $\chi(x)$: function (locally) Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter and $\equiv$: equals by definition Source: Olver (1997b, (13.02), p. 225) Referenced by: Erratum (V1.0.17) for Equations (9.7.3), (9.7.4) Permalink: http://dlmf.nist.gov/9.7.E3 Encodings: TeX, pMML, png Modification (effective with 1.0.17): Originally the function $\chi$ was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$. See also: Annotations for §9.7(i), §9.7 and Ch.9

For large $x$,

 9.7.4 $\chi(x)\sim(\tfrac{1}{2}\pi x)^{\ifrac{1}{2}}.$ ⓘ Symbols: $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter and $\chi(x)$: function Source: Olver (1997b, p. 225) Referenced by: Erratum (V1.0.17) for Equations (9.7.3), (9.7.4) Permalink: http://dlmf.nist.gov/9.7.E4 Encodings: TeX, pMML, png Modification (effective with 1.0.17): Originally the formula was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$. See also: Annotations for §9.7(i), §9.7 and Ch.9

Numerical values of $\chi(n)$ are given in Table 9.7.1 for $n=1(1)20$ to 2D.

## §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 $\chi(n+\sigma)+1$ where $\sigma=\frac{1}{6}$ for (9.7.7) and $\sigma=0$ for (9.7.8), provided that $n\geq 0$ in the first case and $n\geq 1$ in the second case.

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 natural logarithm, $x$: 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 Ch.9
 9.7.16 $\displaystyle\mathrm{Bi}\left(x\right)$ $\displaystyle\leq\frac{{\mathrm{e}}^{\xi}}{\sqrt{\pi}x^{1/4}}\left(1+\left(% \chi(\tfrac{7}{6})+1\right)\frac{5}{72\xi}\right),$ $\displaystyle\mathrm{Bi}'\left(x\right)$ $\displaystyle\leq\frac{x^{1/4}e^{\xi}}{\sqrt{\pi}}\left(1+\left(\frac{\pi}{2}+% 1\right)\frac{7}{72\xi}\right),$ ⓘ 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 natural logarithm, $x$: real variable, $\chi(x)$: function and $\xi$: change of variable Source: Combine Nemes (2017b, (26) and Proposition B.3) with (9.6.4), (9.6.5), and (10.27.8). Referenced by: §9.7(iii), Erratum (V1.0.17) for Subsection 9.7(iii) Permalink: http://dlmf.nist.gov/9.7.E16 Encodings: TeX, TeX, pMML, pMML, png, png Modification (effective with 1.0.17): The bounds on the right-hands sides were sharpened. The factors appearing on the right-hand sides given by $\left(\chi(\tfrac{7}{6})+1\right)\frac{5}{72\xi}$, $\left(\frac{\pi}{2}+1\right)\frac{7}{72\xi}$, originally were $\frac{5\pi}{72\xi}\exp\left(\frac{5\pi}{72\xi}\right)$, $\frac{7\pi}{72\xi}\exp\left(\frac{7\pi}{72\xi}\right)$, respectively. See also: Annotations for §9.7(iii), §9.7 and Ch.9

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

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

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 $\begin{cases}1,&|\operatorname{ph}z|\leq\tfrac{1}{3}\pi,\\ \min\left(|\csc\left(\operatorname{ph}\zeta\right)|,\chi(n+\sigma)+1\right),&% \tfrac{1}{3}\pi\leq|\operatorname{ph}z|\leq\tfrac{2}{3}\pi,\\ \frac{\sqrt{2\pi(n+\sigma)}}{|\cos\left(\operatorname{ph}\zeta\right)|^{n+% \sigma}}+\chi(n+\sigma)+1,&\tfrac{2}{3}\pi\leq|\operatorname{ph}z|<\pi,\end{cases}$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\csc\NVar{z}$: cosecant function, $\cos\NVar{z}$: cosine function, $\exp\NVar{z}$: exponential function, $\operatorname{ph}$: phase, $\Re$: real part, $z$: complex variable, $\zeta$: change of variable, $\chi(x)$: function, $n$: index and $\sigma$: real variable Source: Derivable from (9.6.1)–(9.6.5) and Nemes (2017b, (26) and Propositions B.1, B.3, B.4). Referenced by: §9.7(iv), Erratum (V1.0.11) for Equation (9.7.17) Permalink: http://dlmf.nist.gov/9.7.E17 Encodings: TeX, pMML, png Modification (effective with 1.0.17): The bounds have been sharpened for $|\operatorname{ph}z|\leq\tfrac{1}{3}\pi$, from $2\exp\left(\dfrac{\sigma}{36|\zeta|}\right)$ to $1$; for $\tfrac{1}{3}\pi\leq|\operatorname{ph}z|\leq\tfrac{2}{3}\pi$, from $2\chi(n)\exp\left(\dfrac{\sigma\pi}{72|\zeta|}\right)$ to $\min\left(|\csc\left(\operatorname{ph}\zeta\right)|,\chi(n+\sigma)+1\right)$; and for $\tfrac{2}{3}\pi\leq|\operatorname{ph}z|<\pi,$ from $\dfrac{4\chi(n)}{|\cos\left(\operatorname{ph}\zeta\right)|^{n}}\exp\left(% \dfrac{\sigma\pi}{36|\Re\zeta|}\right)$ to $\frac{\sqrt{2\pi(n+\sigma)}}{|\cos\left(\operatorname{ph}\zeta\right)|^{n+% \sigma}}+\chi(n+\sigma)+1$. 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 Ch.9

provided that $n\geq 0$, $\sigma=\tfrac{1}{6}$ for (9.7.5) and $n\geq 1$, $\sigma=0$ 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.

## §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, $\zeta$: 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 Ch.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, $\zeta$: 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 Ch.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 natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$: 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 Ch.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).