# §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

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

 9.7.2 $\displaystyle u_{k}$ $\displaystyle=\frac{(2k+1)(2k+3)(2k+5)\cdots(6k-1)}{(216)^{k}(k)!},$ $\displaystyle v_{k}$ $\displaystyle=\frac{6k+1}{1-6k}u_{k}.$

Lastly,

 9.7.3 $\chi(n)=\pi^{\ifrac{1}{2}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}n+1% \right)/\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}n+\tfrac{1}{2}\right).$ Defines: $\chi(n)$: function (locally) Symbols: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$: gamma function Permalink: http://dlmf.nist.gov/9.7.E3 Encodings: TeX, pMML, png

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 and $\chi(n)$: function Permalink: http://dlmf.nist.gov/9.7.E4 Encodings: TeX, pMML, png

## §9.7(ii) Poincaré-Type Expansions

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

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

## §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 $2\chi(n)\mathop{\exp\/}\nolimits\left(\sigma\pi/(72\zeta)\right)$ 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 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ $\displaystyle\leq\frac{e^{-\xi}}{2\sqrt{\pi}x^{1/4}}$, $\displaystyle|\mathop{\mathrm{Ai}\/}\nolimits'\!\left(x\right)|$ $\displaystyle\leq\frac{x^{1/4}e^{-\xi}}{2\sqrt{\pi}}\left(1+\frac{7}{72\xi}\right)$,
 9.7.16 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ $\displaystyle\leq\frac{e^{\xi}}{\sqrt{\pi}x^{1/4}}\left(1+\frac{5\pi}{72\xi}% \mathop{\exp\/}\nolimits\left(\frac{5\pi}{72\xi}\right)\right),$ $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits'\!\left(x\right)$ $\displaystyle\leq\frac{x^{1/4}e^{\xi}}{\sqrt{\pi}}\left(1+\frac{7\pi}{72\xi}% \mathop{\exp\/}\nolimits\left(\frac{7\pi}{72\xi}\right)\right),$

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 $2\mathop{\exp\/}\nolimits\!\left(\frac{\sigma}{36|\zeta|}\right)$, $2\chi(n)\mathop{\exp\/}\nolimits\!\left(\frac{\sigma\pi}{72|\zeta|}\right)$or $\frac{4\chi(n)}{|\mathop{\cos\/}\nolimits\!\left(\mathop{\mathrm{ph}\/}% \nolimits\zeta\right)|^{n}}\mathop{\exp\/}\nolimits\!\left(\frac{\sigma\pi}{36% |\Re\zeta|}\right)$,

according as $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{3}\pi$, $\tfrac{1}{3}\pi\leq|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{2}{3}\pi$, or $\tfrac{2}{3}\pi\leq|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$. 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 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{e^{-\zeta}}{2\sqrt{\pi}z^{1/4}}\left(\sum_{k=0}^{n-1}(-1)^% {k}\frac{u_{k}}{\zeta^{k}}+R_{n}(z)\right),$ 9.7.19 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits'\!\left(z\right)$ $\displaystyle=-\frac{z^{1/4}e^{-\zeta}}{2\sqrt{\pi}}\left(\sum_{k=0}^{n-1}(-1)% ^{k}\frac{v_{k}}{\zeta^{k}}+S_{n}(z)\right),$

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

 9.7.20 $\displaystyle R_{n}(z)$ $\displaystyle=(-1)^{n}\sum_{k=0}^{m-1}(-1)^{k}u_{k}\frac{\mathop{G_{n-k}\/}% \nolimits\!\left(2\zeta\right)}{\zeta^{k}}+R_{m,n}(z),$ Defines: $R_{n}$: remainder function (locally) Symbols: $\mathop{G_{p}\/}\nolimits\!\left(z\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: change of variable, $m$: index, $n$: index and $u_{s}$: expansion coefficient Permalink: http://dlmf.nist.gov/9.7.E20 Encodings: TeX, pMML, png 9.7.21 $\displaystyle S_{n}(z)$ $\displaystyle=(-1)^{n-1}\sum_{k=0}^{m-1}(-1)^{k}v_{k}\frac{\mathop{G_{n-k}\/}% \nolimits\!\left(2\zeta\right)}{\zeta^{k}}+S_{m,n}(z),$ Defines: $S_{n}$: remainder function (locally) Symbols: $\mathop{G_{p}\/}\nolimits\!\left(z\right)$: rescaled terminant function, $k$: nonnegative integer, $z$: complex variable, $\zeta(z)$: change of variable, $m$: index, $n$: index and $v_{s}$: expansion coefficient Permalink: http://dlmf.nist.gov/9.7.E21 Encodings: TeX, pMML, png

where

 9.7.22 $\mathop{G_{p}\/}\nolimits\!\left(z\right)=\frac{e^{z}}{2\pi}\mathop{\Gamma\/}% \nolimits\!\left(p\right)\mathop{\Gamma\/}\nolimits\!\left(1-p,z\right).$ Defines: $\mathop{G_{p}\/}\nolimits\!\left(z\right)$: rescaled terminant function Symbols: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$: gamma function, $e$: base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$: incomplete gamma function and $z$: complex variable Permalink: http://dlmf.nist.gov/9.7.E22 Encodings: TeX, pMML, png

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

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

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).