§ 9.7. Asymptotic Expansions

Referenced by:: §9.11(i), §9.12(viii), §9.17(i)
Permalink:: http://dlmf.nist.gov/9.7

§9.7(i)Notation
§9.7(ii)Poincaré-Type Expansions
§9.7(iii)Error Bounds for Real Variables
§9.7(iv)Error Bounds for Complex Variables
§9.7(v)Exponentially-Improved Expansions

§ 9.7(i). Notation

Notes:: See Olver (1997b, p. 225).
Referenced by:: §9.12(viii)
Permalink:: http://dlmf.nist.gov/9.7.SS1

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
Symbols:: $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.7.E1
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Also $u_{0}=v_{0}=1$ and for $k=1,2,\ldots,$

9.7.2

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

$v_{k}=\frac{6k+1}{1-6k}u_{k}.$

Defines:: $u_{s}$ : expansion coefficient and $v_{s}$ : expansion coefficient
Symbols:: $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/9.7.E2
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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
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function
Permalink:: http://dlmf.nist.gov/9.7.E3
Encodings:: TeX, pMathML, 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:: $\sim$ : asymptotically equal and $\chi(n)$ : function
Permalink:: http://dlmf.nist.gov/9.7.E4
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9.7.1. $\chi(n)$ .

$n$	$\chi(n)$	$n$	$\chi(n)$
1	1.57	11	4.25
2	2.00	12	4.43
3	2.36	13	4.61
4	2.67	14	4.77
5	2.95	15	4.94
6	3.20	16	5.09
7	3.44	17	5.24
8	3.66	18	5.39
9	3.87	19	5.54
10	4.06	20	5.68

Symbols:: $\chi(n)$ : function
Referenced by:: §9.7(i)
Permalink:: http://dlmf.nist.gov/9.7.T1

§ 9.7(ii). Poincaré-Type Expansions

Notes:: See Olver (1997b, pp. 392–393 and 413–414).
Referenced by:: §9.10(ii), §9.11(v), §9.12(iv)
Permalink:: http://dlmf.nist.gov/9.7.SS2

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

9.7.5 $\mathrm{Ai}\!\left(z\right)\sim\frac{e^{{-\zeta}}}{2\sqrt{\pi}z^{{1/4}}}\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{k}}{\zeta^{k}},$ $|\mathrm{ph}z|\leq\pi-\delta$ ,

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\sim$ : asymptotically equal, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $u_{s}$ : expansion coefficient
Referenced by:: §9.7(iii), §9.7(iv), §9.7(v)
Permalink:: http://dlmf.nist.gov/9.7.E5
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9.7.6 ${{\mathrm{Ai}}^{{\prime}}}\!\left(z\right)\sim-\frac{z^{{1/4}}e^{{-\zeta}}}{2\sqrt{\pi}}\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{k}}{\zeta^{k}},$ $|\mathrm{ph}z|\leq\pi-\delta$ ,

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\sim$ : asymptotically equal, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $v_{s}$ : expansion coefficient
Referenced by:: §9.7(iii), §9.7(iv), §9.7(v)
Permalink:: http://dlmf.nist.gov/9.7.E6
Encodings:: TeX, pMathML, png

9.7.7 $\mathrm{Bi}\!\left(z\right)\sim\frac{e^{{\zeta}}}{\sqrt{\pi}z^{{1/4}}}\sum _{{k=0}}^{{\infty}}\frac{u_{k}}{\zeta^{k}},$ $|\mathrm{ph}z|\leq\tfrac{1}{3}\pi-\delta$ ,

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\sim$ : asymptotically equal, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $u_{s}$ : expansion coefficient
Referenced by:: §9.7(iii), §9.7(iv), §9.7(v)
Permalink:: http://dlmf.nist.gov/9.7.E7
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9.7.8 ${{\mathrm{Bi}}^{{\prime}}}\!\left(z\right)\sim\frac{z^{{1/4}}e^{{\zeta}}}{\sqrt{\pi}}\sum _{{k=0}}^{{\infty}}\frac{v_{k}}{\zeta^{k}},$ $|\mathrm{ph}z|\leq\tfrac{1}{3}\pi-\delta$ .

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\sim$ : asymptotically equal, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $v_{s}$ : expansion coefficient
Referenced by:: §9.7(iii)
Permalink:: http://dlmf.nist.gov/9.7.E8
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9.7.9 $\mathrm{Ai}\!\left(-z\right)\sim\frac{1}{\sqrt{\pi}z^{{1/4}}}\left(\cos\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k}}}{\zeta^{{2k}}}+\sin\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathrm{ph}z|\leq\tfrac{2}{3}\pi-\delta$ ,

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\sim$ : asymptotically equal, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $u_{s}$ : expansion coefficient
Referenced by:: §9.7(iii)
Permalink:: http://dlmf.nist.gov/9.7.E9
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9.7.10 ${{\mathrm{Ai}}^{{\prime}}}\!\left(-z\right)\sim\frac{z^{{1/4}}}{\sqrt{\pi}}\left(\sin\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k}}}{\zeta^{{2k}}}-\cos\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathrm{ph}z|\leq\tfrac{2}{3}\pi-\delta$ ,

Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $\sim$ : asymptotically equal, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $v_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E10
Encodings:: TeX, pMathML, png

9.7.11 $\mathrm{Bi}\!\left(-z\right)\sim\frac{1}{\sqrt{\pi}z^{{1/4}}}\left(-\sin\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k}}}{\zeta^{{2k}}}+\cos\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathrm{ph}z|\leq\tfrac{2}{3}\pi-\delta$ ,

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\sim$ : asymptotically equal, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $u_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E11
Encodings:: TeX, pMathML, png

9.7.12 ${{\mathrm{Bi}}^{{\prime}}}\!\left(-z\right)\sim\frac{z^{{1/4}}}{\sqrt{\pi}}\left(\cos\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k}}}{\zeta^{{2k}}}+\sin\!\left(\zeta-\tfrac{1}{4}\pi\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathrm{ph}z|\leq\tfrac{2}{3}\pi-\delta$ .

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\sim$ : asymptotically equal, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $v_{s}$ : expansion coefficient
Referenced by:: §9.7(iii)
Permalink:: http://dlmf.nist.gov/9.7.E12
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9.7.13 $\mathrm{Bi}\!\left(ze^{{\pm\pi i/3}}\right)\mathrel{\sim}\sqrt{\frac{2}{\pi}}\frac{e^{{\pm\pi i/6}}}{z^{{1/4}}}\*\left(\cos\!\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}i\ln 2\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k}}}{\zeta^{{2k}}}+\sin\!\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}i\ln 2\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{u_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathrm{ph}z|\leq\tfrac{2}{3}\pi-\delta$ ,

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\sim$ : asymptotically equal, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $u_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E13
Encodings:: TeX, pMathML, png

9.7.14 ${{\mathrm{Bi}}^{{\prime}}}\!\left(ze^{{\pm\pi i/3}}\right)\mathrel{\sim}\sqrt{\frac{2}{\pi}}e^{{\mp\pi i/6}}z^{{1/4}}\*\left(-\sin\!\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}i\ln 2\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k}}}{\zeta^{{2k}}}+\cos\!\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}i\ln 2\right)\sum _{{k=0}}^{{\infty}}(-1)^{k}\frac{v_{{2k+1}}}{\zeta^{{2k+1}}}\right),$ $|\mathrm{ph}z|\leq\tfrac{2}{3}\pi-\delta$ .

Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function, $\sim$ : asymptotically equal, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta(z)$ : change of variable and $v_{s}$ : expansion coefficient
Referenced by:: §9.7(iv), §9.7(v)
Permalink:: http://dlmf.nist.gov/9.7.E14
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§ 9.7(iii). Error Bounds for Real Variables

Notes:: See Olver (1997b, pp. 392–393 and 413–414).
Permalink:: http://dlmf.nist.gov/9.7.SS3

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 $0<x<\infty$

9.7.15

$\mathrm{Ai}\!\left(x\right)\leq\frac{e^{{-\xi}}}{2\sqrt{\pi}x^{{1/4}}}$ ,

$|{{\mathrm{Ai}}^{{\prime}}}\!\left(x\right)|\leq\frac{x^{{1/4}}e^{{-\xi}}}{2\sqrt{\pi}}\left(1+\frac{7}{72\xi}\right)$ ,

Defines:: $xi$ : change of variable
Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.7.E15
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9.7.16

$\mathrm{Bi}\!\left(x\right)\leq\frac{e^{{\xi}}}{\sqrt{\pi}x^{{1/4}}}\left(1+\frac{5\pi}{72\xi}\exp\left(\frac{5\pi}{72\xi}\right)\right),$

${{\mathrm{Bi}}^{{\prime}}}\!\left(x\right)\leq\frac{x^{{1/4}}e^{{\xi}}}{\sqrt{\pi}}\left(1+\frac{7\pi}{72\xi}\exp\left(\frac{7\pi}{72\xi}\right)\right),$

Defines:: $xi$ : change of variable
Symbols:: $\mathrm{Bi}\!\left(z\right)$ : Airy function and $x$ : real variable
Permalink:: http://dlmf.nist.gov/9.7.E16
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where $\xi=\tfrac{2}{3}x^{{3/2}}$ .

§ 9.7(iv). Error Bounds for Complex Variables

Notes:: These results can be derived from (9.6.1)–(9.6.5) and Olver (1997b, pp. 266–267).
Referenced by:: §9.7(v)
Permalink:: http://dlmf.nist.gov/9.7.SS4

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

$2\exp\!\left(\frac{\sigma}{36|\zeta|}\right)$ ,

$2\chi(n)\exp\!\left(\frac{\sigma\pi}{72|\zeta|}\right)$ or

$\frac{4\chi(n)}{|\cos\!\left(\mathrm{ph}\zeta\right)|^{n}}\exp\!\left(\frac{\sigma\pi}{36|\Re\zeta|}\right)$ ,

Defines:: $n$ : index and $\sigma$ : index
Symbols:: $\zeta(z)$ : change of variable and $\chi(n)$ : function
Permalink:: http://dlmf.nist.gov/9.7.E17
Encodings:: TeX, TeX, TeX, pMathML, pMathML, pMathML, png, png, png

according as $|\mathrm{ph}z|\leq\tfrac{1}{3}\pi$ , $\tfrac{1}{3}\pi\leq|\mathrm{ph}z|\leq\tfrac{2}{3}\pi$ , or $\tfrac{2}{3}\pi\leq|\mathrm{ph}z|\leq\pi$ . 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

Notes:: See Olver (1991a, 1993).
Referenced by:: §9.17(i)
Permalink:: http://dlmf.nist.gov/9.7.SS5

In (9.7.5) and (9.7.6) let

9.7.18 $\mathrm{Ai}\!\left(z\right)=\frac{e^{{-\zeta}}}{2\sqrt{\pi}z^{{1/4}}}\left(\sum _{{k=0}}^{{n-1}}(-1)^{k}\frac{u_{k}}{\zeta^{k}}+R_{n}(z)\right),$

Defines:: $n$ : index
Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : change of variable, $R_{n}$ : remainder function and $u_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E18
Encodings:: TeX, pMathML, png

9.7.19 ${{\mathrm{Ai}}^{{\prime}}}\!\left(z\right)=-\frac{z^{{1/4}}e^{{-\zeta}}}{2\sqrt{\pi}}\left(\sum _{{k=0}}^{{n-1}}(-1)^{k}\frac{v_{k}}{\zeta^{k}}+S_{n}(z)\right),$

Defines:: $n$ : index
Symbols:: $\mathrm{Ai}\!\left(z\right)$ : Airy function, $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : change of variable, $S_{n}$ : remainder function and $v_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E19
Encodings:: TeX, pMathML, png

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

9.7.20 $R_{n}(z)=(-1)^{n}\sum _{{k=0}}^{{m-1}}(-1)^{k}u_{k}\frac{G_{{n-k}}(2\zeta)}{\zeta^{k}}+R_{{m,n}}(z),$

Defines:: $m$ : index, $n$ : index and $R_{n}$ : remainder function
Symbols:: $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : change of variable, $G_{p}$ : function and $u_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E20
Encodings:: TeX, pMathML, png

9.7.21 $S_{n}(z)=(-1)^{{n-1}}\sum _{{k=0}}^{{m-1}}(-1)^{k}v_{k}\frac{G_{{n-k}}(2\zeta)}{\zeta^{k}}+S_{{m,n}}(z),$

Defines:: $m$ : index, $n$ : index and $S_{n}$ : remainder function
Symbols:: $k$ : nonnegative integer, $z$ : complex variable, $\zeta(z)$ : change of variable, $G_{p}$ : function and $v_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.7.E21
Encodings:: TeX, pMathML, png

where

9.7.22 $G_{p}(z)=\frac{e^{z}}{2\pi}\Gamma\!\left(p\right)\Gamma\!\left(1-p,z\right).$

Defines:: $G_{p}$ : function
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.7.E22
Encodings:: TeX, pMathML, png

(For the notation see §Ch.8.) And as $z\rightarrow\infty$ with $m$ fixed

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

Defines:: $m$ : index and $n$ : index
Symbols:: $O\!\left(x\right)$ : order symbol, $z$ : complex variable, $\zeta(z)$ : change of variable, $R_{n}$ : remainder function and $S_{n}$ : remainder function
Permalink:: http://dlmf.nist.gov/9.7.E23
Encodings:: TeX, pMathML, png

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).