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DLMF: §9.5 Integral Representations ‣ Airy Functions ‣ Chapter 9 Airy and Related Functions

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9 Airy and Related Functions Airy Functions 9.4 Maclaurin Series 9.6 Relations to Other Functions

§9.5 Integral Representations

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Permalink:: http://dlmf.nist.gov/9.5
See also:: Annotations for Ch.9

Contents

§9.5(i) Real Variable

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Keywords:: Airy functions, integral representations
Notes:: See Olver (1997b, pp. 53 and 103).
Permalink:: http://dlmf.nist.gov/9.5.i
See also:: Annotations for §9.5 and Ch.9

9.5.1		$\operatorname{Ai}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\cos\left(\tfrac% {1}{3}t^{3}+xt\right)\,\mathrm{d}t.$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $x$ : real variable Source: Olver (1997b, p. 53) A&S Ref: 10.4.32 (in different form) Permalink: http://dlmf.nist.gov/9.5.E1 Encodings: TeX, pMML, png See also: Annotations for §9.5(i), §9.5 and Ch.9

9.5.2		$\operatorname{Ai}\left(-x\right)=\frac{x^{\ifrac{1}{2}}}{\pi}\int_{-1}^{\infty% }\cos\left(x^{\ifrac{3}{2}}(\tfrac{1}{3}t^{3}+t^{2}-\tfrac{2}{3})\right)\,% \mathrm{d}t,$
$x>0$ .
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $x$ : real variable Source: Olver (1997b, p. 103) A&S Ref: 10.4.32 (in different form) Permalink: http://dlmf.nist.gov/9.5.E2 Encodings: TeX, pMML, png See also: Annotations for §9.5(i), §9.5 and Ch.9

9.5.3		$\operatorname{Bi}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-{% \tfrac{1}{3}}t^{3}+xt\right)\,\mathrm{d}t+\frac{1}{\pi}\int_{0}^{\infty}\sin% \left(\tfrac{1}{3}t^{3}+xt\right)\,\mathrm{d}t.$
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b})$ : half-closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b}]$ : half-closed interval, $\sin\NVar{z}$ : sine function and $x$ : real variable Proof sketch: Use (9.5.5) with the substitions: for the paths $(-\infty,0]$ use $t=-\tau$ , for the paths $[0,\infty{\mathrm{e}}^{\pm\pi\mathrm{i}/3})$ use $t={\mathrm{e}}^{\pm\pi\mathrm{i}/3}\tau$ . For the resulting integrals use (4.14.1) and (4.14.2). A&S Ref: 10.4.33 Referenced by: (9.12.19) Permalink: http://dlmf.nist.gov/9.5.E3 Encodings: TeX, pMML, png See also: Annotations for §9.5(i), §9.5 and Ch.9

See also (9.10.19), (9.11.3), (36.9.2), and Vallée and Soares (2010, §2.1.3).

§9.5(ii) Complex Variable

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Keywords:: Airy functions, integral representations
Notes:: See Olver (1997b, p. 53), Reid (1995), Copson (1963).
Permalink:: http://dlmf.nist.gov/9.5.ii
See also:: Annotations for §9.5 and Ch.9

9.5.4		$\operatorname{Ai}\left(z\right)=\frac{1}{2\pi i}\int_{\infty e^{-\pi i/3}}^{% \infty e^{\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\,\mathrm{d}t,$
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $z$ : complex variable Source: Olver (1997b, p. 53) Referenced by: §36.2(ii), §9.14, §9.17(iii), (9.5.5) Permalink: http://dlmf.nist.gov/9.5.E4 Encodings: TeX, pMML, png See also: Annotations for §9.5(ii), §9.5 and Ch.9

9.5.5		$\operatorname{Bi}\left(z\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty e^{\pi i/% 3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\,\mathrm{d}t+\dfrac{1}{2\pi}\int_{-% \infty}^{\infty e^{-\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\,\mathrm{d}t.$
ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $z$ : complex variable Proof sketch: Combine (9.2.10) and (9.5.4). Referenced by: (9.5.3) Permalink: http://dlmf.nist.gov/9.5.E5 Encodings: TeX, pMML, png See also: Annotations for §9.5(ii), §9.5 and Ch.9

9.5.6		$\operatorname{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi}\int_{0}^{\infty}\exp% \left(-\frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}\right)\,\mathrm{d}t,$
$\|\operatorname{ph}z\|<\tfrac{1}{6}\pi$ .
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\operatorname{ph}$ : phase and $z$ : complex variable Source: Reid (1995, (5.4), p. 170, with change of variable and using analytic continuation to complex $z$ ) Referenced by: Erratum (V1.0.15) for Equation (9.5.6) Permalink: http://dlmf.nist.gov/9.5.E6 Encodings: TeX, pMML, png Clarification (effective with 1.0.15): The validity constraint $\|\operatorname{ph}z\|<\tfrac{1}{6}\pi$ was added. See also: Annotations for §9.5(ii), §9.5 and Ch.9

9.5.7		$\operatorname{Ai}\left(z\right)=\frac{e^{-\zeta}}{\pi}\int_{0}^{\infty}\exp% \left(-z^{\ifrac{1}{2}}t^{2}\right)\cos\left(\tfrac{1}{3}t^{3}\right)\,\mathrm% {d}t,$
$\|\operatorname{ph}z\|<\pi$ .
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable and $\zeta$ : change of variable Source: Copson (1963, (2.1)) Referenced by: §9.5(ii) Permalink: http://dlmf.nist.gov/9.5.E7 Encodings: TeX, pMML, png See also: Annotations for §9.5(ii), §9.5 and Ch.9

9.5.8		$\operatorname{Ai}\left(z\right)=\frac{e^{-\zeta}\zeta^{\ifrac{-1}{6}}}{\sqrt{% \pi}(48)^{\ifrac{1}{6}}\Gamma\left(\frac{5}{6}\right)}\int_{0}^{\infty}e^{-t}t% ^{-\ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-\ifrac{1}{6}}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|<\frac{2}{3}\pi$ .
ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable and $\zeta$ : change of variable Proof sketch: Follows from (9.6.21) and (13.4.4). Referenced by: §9.17(iii), §9.5(ii) Permalink: http://dlmf.nist.gov/9.5.E8 Encodings: TeX, pMML, png See also: Annotations for §9.5(ii), §9.5 and Ch.9

In (9.5.7) and (9.5.8) $\zeta=\frac{2}{3}z^{\ifrac{3}{2}}$ .

See also (9.10.18) and (9.11.4).

9.4 Maclaurin Series 9.6 Relations to Other Functions

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