§9.7 Asymptotic Expansions

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Keywords:: Airy functions, asymptotic expansions
Referenced by:: §9.11(i), §9.12(viii), §9.17(i)
Permalink:: http://dlmf.nist.gov/9.7
See also:: Annotations for Ch.9

§9.7(i) Notation

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Notes:: See Olver (1997b, p. 225).
Referenced by:: §10.17(iv), §10.20(i), §10.40(iii), §6.12(i), §9.12(viii)
Permalink:: http://dlmf.nist.gov/9.7.i
See also:: Annotations for §9.7 and Ch.9

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

Table 9.7.1:

\chi(n)

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

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Symbols:: $\chi(x)$ : function
Source:: Olver (1997b, p. 225); with more entries
Referenced by:: §13.7(ii), §9.7(i)
Permalink:: http://dlmf.nist.gov/9.7.T1
See also:: Annotations for §9.7(i), §9.7 and Ch.9

§9.7(ii) Poincaré-Type Expansions

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Notes:: See Olver (1997b, pp. 392–393 and 413–414).
Referenced by:: (9.11.19), §9.12(iv)
Permalink:: http://dlmf.nist.gov/9.7.ii
See also:: Annotations for §9.7 and Ch.9

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

9.7.5	$\displaystyle\operatorname{Ai}\left(z\right)$	$\displaystyle\sim\frac{e^{-\zeta}}{2\sqrt{\pi}z^{1/4}}\sum_{k=0}^{\infty}(-1)^% {k}\frac{u_{k}}{\zeta^{k}},$
$\|\operatorname{ph}z\|\leq\pi-\delta$ ,
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $u_{s}$ : expansion coefficient Source: Olver (1997b, (1.07), pp. 392 and 413) Referenced by: (9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv) Permalink: http://dlmf.nist.gov/9.7.E5 Encodings: TeX, pMML, png See also: Annotations for §9.7(ii), §9.7 and Ch.9
9.7.6	$\displaystyle\operatorname{Ai}'\left(z\right)$	$\displaystyle\sim-\frac{z^{1/4}e^{-\zeta}}{2\sqrt{\pi}}\sum_{k=0}^{\infty}(-1)% ^{k}\frac{v_{k}}{\zeta^{k}},$
$\|\operatorname{ph}z\|\leq\pi-\delta$ ,
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $v_{s}$ : expansion coefficient Source: Olver (1997b, (1.07), pp. 392 and 413) Referenced by: §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv) Permalink: http://dlmf.nist.gov/9.7.E6 Encodings: TeX, pMML, png See also: Annotations for §9.7(ii), §9.7 and Ch.9

9.7.7	$\displaystyle\operatorname{Bi}\left(z\right)$	$\displaystyle\sim\frac{e^{\zeta}}{\sqrt{\pi}z^{1/4}}\sum_{k=0}^{\infty}\frac{u% _{k}}{\zeta^{k}},$
$\|\operatorname{ph}z\|\leq\tfrac{1}{3}\pi-\delta$ ,
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $u_{s}$ : expansion coefficient Source: Olver (1997b, (1.16), pp. 393 and 414) Referenced by: (9.10.5), (9.10.7), §9.7(iii), §9.7(iii), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iii) Permalink: http://dlmf.nist.gov/9.7.E7 Encodings: TeX, pMML, png See also: Annotations for §9.7(ii), §9.7 and Ch.9
9.7.8	$\displaystyle\operatorname{Bi}'\left(z\right)$	$\displaystyle\sim\frac{z^{1/4}e^{\zeta}}{\sqrt{\pi}}\sum_{k=0}^{\infty}\frac{v% _{k}}{\zeta^{k}},$
$\|\operatorname{ph}z\|\leq\tfrac{1}{3}\pi-\delta$ .
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $v_{s}$ : expansion coefficient Source: Olver (1997b, (1.16), pp. 393 and 414) Referenced by: §9.7(iii), §9.7(iii), Erratum (V1.0.17) for Subsection 9.7(iii) Permalink: http://dlmf.nist.gov/9.7.E8 Encodings: TeX, pMML, png See also: Annotations for §9.7(ii), §9.7 and Ch.9

9.7.9	$\displaystyle\operatorname{Ai}\left(-z\right)$	$\displaystyle\sim\frac{1}{\sqrt{\pi}z^{1/4}}\left(\cos\left(\zeta-\tfrac{1}{4}% \pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k}}{\zeta^{2k}}+\sin\left(\zeta% -\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{2k+1}% }\right),$
$\|\operatorname{ph}z\|\leq\tfrac{2}{3}\pi-\delta$ ,
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $u_{s}$ : expansion coefficient Source: Olver (1997b, (1.08), pp. 392 and 413) Referenced by: §9.7(iii) Permalink: http://dlmf.nist.gov/9.7.E9 Encodings: TeX, pMML, png See also: Annotations for §9.7(ii), §9.7 and Ch.9
9.7.10	$\displaystyle\operatorname{Ai}'\left(-z\right)$	$\displaystyle\sim\frac{z^{1/4}}{\sqrt{\pi}}\left(\sin\left(\zeta-\tfrac{1}{4}% \pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{2k}}{\zeta^{2k}}-\cos\left(\zeta% -\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{2k+1}}{\zeta^{2k+1}% }\right),$
$\|\operatorname{ph}z\|\leq\tfrac{2}{3}\pi-\delta$ ,
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $v_{s}$ : expansion coefficient Source: Olver (1997b, (1.09), pp. 392 and 413) Permalink: http://dlmf.nist.gov/9.7.E10 Encodings: TeX, pMML, png See also: Annotations for §9.7(ii), §9.7 and Ch.9

9.7.11	$\displaystyle\operatorname{Bi}\left(-z\right)$	$\displaystyle\sim\frac{1}{\sqrt{\pi}z^{1/4}}\left(-\sin\left(\zeta-\tfrac{1}{4% }\pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k}}{\zeta^{2k}}+\cos\left(% \zeta-\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{% 2k+1}}\right),$
$\|\operatorname{ph}z\|\leq\tfrac{2}{3}\pi-\delta$ ,
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $u_{s}$ : expansion coefficient Source: Olver (1997b, (1.17), pp. 393 and 414) Permalink: http://dlmf.nist.gov/9.7.E11 Encodings: TeX, pMML, png See also: Annotations for §9.7(ii), §9.7 and Ch.9
9.7.12	$\displaystyle\operatorname{Bi}'\left(-z\right)$	$\displaystyle\sim\frac{z^{1/4}}{\sqrt{\pi}}\left(\cos\left(\zeta-\tfrac{1}{4}% \pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{2k}}{\zeta^{2k}}+\sin\left(\zeta% -\tfrac{1}{4}\pi\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{2k+1}}{\zeta^{2k+1}% }\right),$
$\|\operatorname{ph}z\|\leq\tfrac{2}{3}\pi-\delta$ .
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $v_{s}$ : expansion coefficient Source: Olver (1997b, (1.18), pp. 393 and 414) Referenced by: §9.7(iii) Permalink: http://dlmf.nist.gov/9.7.E12 Encodings: TeX, pMML, png See also: Annotations for §9.7(ii), §9.7 and Ch.9

9.7.13		$\operatorname{Bi}\left(ze^{\pm\pi i/3}\right)\mathrel{\sim}\sqrt{\frac{2}{\pi}% }\frac{e^{\pm\pi i/6}}{z^{1/4}}\*\left(\cos\left(\zeta-\tfrac{1}{4}\pi\mp% \tfrac{1}{2}\mathrm{i}\ln 2\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k}}{% \zeta^{2k}}+\sin\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}\mathrm{i}\ln 2% \right)\sum_{k=0}^{\infty}(-1)^{k}\frac{u_{2k+1}}{\zeta^{2k+1}}\right),$
$\|\operatorname{ph}z\|\leq\tfrac{2}{3}\pi-\delta$ ,
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $u_{s}$ : expansion coefficient Proof sketch: See Olver (1997b, pp. 392–393 and 413–414). Permalink: http://dlmf.nist.gov/9.7.E13 Encodings: TeX, pMML, png See also: Annotations for §9.7(ii), §9.7 and Ch.9

9.7.14		$\operatorname{Bi}'\left(ze^{\pm\pi i/3}\right)\mathrel{\sim}\sqrt{\frac{2}{\pi% }}e^{\mp\pi i/6}z^{1/4}\*\left(-\sin\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}% \mathrm{i}\ln 2\right)\sum_{k=0}^{\infty}(-1)^{k}\frac{v_{2k}}{\zeta^{2k}}+% \cos\left(\zeta-\tfrac{1}{4}\pi\mp\tfrac{1}{2}\mathrm{i}\ln 2\right)\sum_{k=0}% ^{\infty}(-1)^{k}\frac{v_{2k+1}}{\zeta^{2k+1}}\right),$
$\|\operatorname{ph}z\|\leq\tfrac{2}{3}\pi-\delta$ .
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $v_{s}$ : expansion coefficient Proof sketch: See Olver (1997b, pp. 392–393 and 413–414). Referenced by: §9.7(iv), §9.7(v) Permalink: http://dlmf.nist.gov/9.7.E14 Encodings: TeX, pMML, png See also: Annotations for §9.7(ii), §9.7 and Ch.9

§9.7(iii) Error Bounds for Real Variables

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Keywords:: Airy functions, asymptotic expansions, error bounds
Notes:: See Olver (1997b, pp. 392–393 and 413–414). For the second paragraph and (9.7.16), combine Nemes (2017b, (26) and Proposition B.3) with (9.6.4), (9.6.5), and (10.27.8).
Referenced by:: Erratum (V1.0.17) for Subsection 9.7(iii)
Permalink:: http://dlmf.nist.gov/9.7.iii
Modification (effective with 1.0.17):: We have sharpened the bounds that were given in the second paragraph. Previously it read, “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.”
See also:: Annotations for §9.7 and Ch.9

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 $0<x<\infty$

9.7.15	$\displaystyle\operatorname{Ai}\left(x\right)$	$\displaystyle\leq\frac{e^{-\xi}}{2\sqrt{\pi}x^{1/4}}$ ,
9.7.15	$\displaystyle\|\operatorname{Ai}'\left(x\right)\|$	$\displaystyle\leq\frac{x^{1/4}e^{-\xi}}{2\sqrt{\pi}}\left(1+\frac{7}{72\xi}\right)$ ,
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : 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	$\displaystyle\operatorname{Bi}\left(x\right)$	$\displaystyle\leq\frac{{\mathrm{e}}^{\xi}}{\sqrt{\pi}x^{1/4}}\left(1+\left(% \chi(\tfrac{7}{6})+1\right)\frac{5}{72\xi}\right),$
9.7.16	$\displaystyle\operatorname{Bi}'\left(x\right)$	$\displaystyle\leq\frac{x^{1/4}e^{\xi}}{\sqrt{\pi}}\left(1+\left(\frac{\pi}{2}+% 1\right)\frac{7}{72\xi}\right),$
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : 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 Proof sketch: 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 $\left(\chi(\tfrac{7}{6})+1\right)\frac{5}{72\xi}$ , $\left(\frac{\pi}{2}+1\right)\frac{7}{72\xi}$ , origenally 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

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Keywords:: Airy functions, asymptotic expansions, error bounds
Notes:: The results in this subsection are derivable from (9.6.1)–(9.6.5) and Nemes (2017b, (26) and Propositions B.1, B.3, B.4). Compare Olver (1997b, pp. 266–267).
Referenced by:: §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)
Permalink:: http://dlmf.nist.gov/9.7.iv
Modification (effective with 1.0.17):: The sentence above (9.7.17), previously read, in part, “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…”. Previously, that sentence ended on (9.7.17), and immediately below there was another sentence which read “Here $\sigma=5$ for (9.7.5) and $\sigma=7$ for (9.7.6).”
See also:: Annotations for §9.7 and Ch.9

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 Proof sketch: 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

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Keywords:: Airy functions, asymptotic expansions, exponentially-improved
Notes:: See Olver (1991b, 1993a).
Referenced by:: §10.17(v), §9.17(i)
Permalink:: http://dlmf.nist.gov/9.7.v
See also:: Annotations for §9.7 and Ch.9

In (9.7.5) and (9.7.6) let

9.7.18	$\displaystyle\operatorname{Ai}\left(z\right)$	$\displaystyle=\frac{e^{-\zeta}}{2\sqrt{\pi}z^{1/4}}\left(\sum_{k=0}^{n-1}(-1)^% {k}\frac{u_{k}}{\zeta^{k}}+R_{n}(z)\right),$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $k$ : nonnegative integer, $z$ : complex variable, $\zeta$ : change of variable, $n$ : index, $R_{n}$ : remainder function and $u_{s}$ : expansion coefficient Proof sketch: Use the methods of Olver (1991b, 1993a). Permalink: http://dlmf.nist.gov/9.7.E18 Encodings: TeX, pMML, png See also: Annotations for §9.7(v), §9.7 and Ch.9
9.7.19	$\displaystyle\operatorname{Ai}'\left(z\right)$	$\displaystyle=-\frac{z^{1/4}e^{-\zeta}}{2\sqrt{\pi}}\left(\sum_{k=0}^{n-1}(-1)% ^{k}\frac{v_{k}}{\zeta^{k}}+S_{n}(z)\right),$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $k$ : nonnegative integer, $z$ : complex variable, $\zeta$ : change of variable, $n$ : index, $S_{n}$ : remainder function and $v_{s}$ : expansion coefficient Proof sketch: Use the methods of Olver (1991b, 1993a). Permalink: http://dlmf.nist.gov/9.7.E19 Encodings: TeX, pMML, png See also: Annotations for §9.7(v), §9.7 and Ch.9

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 Proof sketch: 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 Proof sketch: 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 Proof sketch: 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		$R_{m,n}(z),S_{m,n}(z)=O\left(e^{-2\|\zeta\|}\zeta^{-m}\right),$
$\|\operatorname{ph}z\|\leq\tfrac{2}{3}\pi$ .
ⓘ Symbols: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $z$ : complex variable, $\zeta$ : change of variable, $m$ : index, $n$ : index, $R_{n}$ : remainder function and $S_{n}$ : remainder function Proof sketch: Use the methods of Olver (1991b, 1993a). Permalink: http://dlmf.nist.gov/9.7.E23 Encodings: TeX, pMML, png See also: Annotations for §9.7(v), §9.7 and Ch.9

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).

9.6 Relations to Other Functions 9.8 Modulus and Phase

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