§9.10 Integrals

9.10.1		$${\int }_z^{\mathrm{\infty }}\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)\right),$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral and $z$: complex variable Source: Combine (9.12.4) and its differentiated form with the first of (9.10.11) Referenced by: (9.10.2) Permalink: http://dlmf.nist.gov/9.10.E1 Encodings: TeX, pMML, png See also: Annotations for §9.10(i), §9.10 and Ch.9

9.10.2		$${\int }_{-\mathrm{\infty }}^z\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right),$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral and $z$: complex variable Source: Combine (9.12.11) and its differentiated form with (9.10.1), then apply (9.2.7) and (9.10.11) Permalink: http://dlmf.nist.gov/9.10.E2 Encodings: TeX, pMML, png See also: Annotations for §9.10(i), §9.10 and Ch.9

9.10.3		$$\begin{array}{ll}{\int }_{-\mathrm{\infty }}^z\mathrm{Bi}\left(t\right)dt& ={\int }_0^z\mathrm{Bi}\left(t\right)dt=\pi \left({\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)-\mathrm{Bi}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)\right)\\ & =\pi \left(\mathrm{Bi}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right).\end{array}$$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral and $z$: complex variable Source: Combine (9.12.5) and its differentiated form with (9.2.7). Permalink: http://dlmf.nist.gov/9.10.E3 Encodings: TeX, pMML, png See also: Annotations for §9.10(i), §9.10 and Ch.9

For the functions $\mathrm{Gi}$ and $\mathrm{Hi}$ see §9.12.

9.10.4		${\displaystyle {\int }_x^{\mathrm{\infty }}}\mathrm{Ai}\left(t\right)dt$	$\sim \frac{1}{2}{\pi }^{-1/2}{x}^{-3/4}\exp \left(-\frac{2}{3}{x}^{3/2}\right),$
$x\to \mathrm{\infty }$,
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\exp z$: exponential function, $\int$: integral and $x$: real variable Source: Integrate the leading term of (9.7.5). Referenced by: §9.10(ii) Permalink: http://dlmf.nist.gov/9.10.E4 Encodings: TeX, pMML, png See also: Annotations for §9.10(ii), §9.10 and Ch.9
9.10.5		${\displaystyle {\int }_0^x}\mathrm{Bi}\left(t\right)dt$	$\sim {\pi }^{-1/2}{x}^{-3/4}\exp \left(\frac{2}{3}{x}^{3/2}\right),$
$x\to \mathrm{\infty }$.
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\exp z$: exponential function, $\int$: integral and $x$: real variable Source: Integrate the leading term of (9.7.7). Permalink: http://dlmf.nist.gov/9.10.E5 Encodings: TeX, pMML, png See also: Annotations for §9.10(ii), §9.10 and Ch.9

9.10.6		$${\int }_{-\mathrm{\infty }}^x\mathrm{Ai}\left(t\right)dt={\pi }^{-1/2}{(-x)}^{-3/4}\cos \left(\frac{2}{3}{(-x)}^{3/2}+\frac{1}{4}\pi \right)+O\left({|x|}^{-9/4}\right),$$
$x\to -\mathrm{\infty }$,
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral and $x$: real variable Source: Integrate the leading terms of (9.7.5) and use (9.10.11) Permalink: http://dlmf.nist.gov/9.10.E6 Encodings: TeX, pMML, png See also: Annotations for §9.10(ii), §9.10 and Ch.9

9.10.7		$${\int }_{-\mathrm{\infty }}^x\mathrm{Bi}\left(t\right)dt={\pi }^{-1/2}{(-x)}^{-3/4}\sin \left(\frac{2}{3}{(-x)}^{3/2}+\frac{1}{4}\pi \right)+O\left({|x|}^{-9/4}\right),$$
$x\to -\mathrm{\infty }$.
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function and $x$: real variable Source: Integrate the leading terms of (9.7.7) and use (9.10.12) Referenced by: §9.10(ii) Permalink: http://dlmf.nist.gov/9.10.E7 Encodings: TeX, pMML, png See also: Annotations for §9.10(ii), §9.10 and Ch.9

For higher terms in (9.10.4)–(9.10.7) see Vallée and Soares (2010, §3.1.3). For error bounds see Boyd (1993).

See also Muldoon (1970).

Let $w(z)$ be any solution of Airy’s equation (9.2.1). Then

9.10.8		${\displaystyle \int zw(z)dz}$	$=w^{\prime }(z),$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $z$: complex variable and $w$: ODE solution Source: To verify, differentiate and refer to (9.2.1). Permalink: http://dlmf.nist.gov/9.10.E8 Encodings: TeX, pMML, png See also: Annotations for §9.10(iii), §9.10 and Ch.9
9.10.9		${\displaystyle \int z^{2}w(z)dz}$	$=z{w}^{\prime }(z)-w(z),$
ⓘ Symbols: $dx$: differential of $x$, $\int$: integral, $z$: complex variable and $w$: ODE solution Source: To verify, differentiate and refer to (9.2.1). Permalink: http://dlmf.nist.gov/9.10.E9 Encodings: TeX, pMML, png See also: Annotations for §9.10(iii), §9.10 and Ch.9

9.10.10		$$\int z^{n+3}w(z)dz=z^{n+2}{w}^{\prime }(z)-(n+2)z^{n+1}w(z)+(n+1)(n+2)\int z^{n}w(z)dz,$$
$n=0,1,2,\mathrm{\dots }.$
ⓘ Defines: $n$: index (locally) Symbols: $dx$: differential of $x$, $\int$: integral, $z$: complex variable and $w$: ODE solution Source: To verify, differentiate and refer to (9.2.1). Permalink: http://dlmf.nist.gov/9.10.E10 Encodings: TeX, pMML, png See also: Annotations for §9.10(iii), §9.10 and Ch.9

See also §9.11(iv).

9.10.11		${\displaystyle {\int }_0^{\mathrm{\infty }}}\mathrm{Ai}\left(t\right)dt$	$=\frac{1}{3}$,
		${\displaystyle {\int }_{-\mathrm{\infty }}^0}\mathrm{Ai}\left(t\right)dt$	$=\frac{2}{3}$,
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$ and $\int$: integral Source: Olver (1997b, p. 431) Referenced by: (9.10.1), (9.10.2), (9.10.6) Permalink: http://dlmf.nist.gov/9.10.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §9.10(iv), §9.10 and Ch.9

9.10.12		$${\int }_{-\mathrm{\infty }}^0\mathrm{Bi}\left(t\right)dt=0.$$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $dx$: differential of $x$ and $\int$: integral Source: Olver (1997b, p. 431) Referenced by: (9.10.7) Permalink: http://dlmf.nist.gov/9.10.E12 Encodings: TeX, pMML, png See also: Annotations for §9.10(iv), §9.10 and Ch.9

9.10.13		$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\mathrm{e}}^{pt}\mathrm{Ai}\left(t\right)dt={\mathrm{e}}^{p^3/3},$$
$\mathrm{\Re }p>0$.
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter Source: Widder (1979, (3.1)) Permalink: http://dlmf.nist.gov/9.10.E13 Encodings: TeX, pMML, png See also: Annotations for §9.10(v), §9.10 and Ch.9

9.10.14		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(t\right)dt={\mathrm{e}}^{-p^3/3}\left(\frac{1}{3}-\frac{p{}_{1}F_1(\frac{1}{3};\frac{4}{3};\frac{1}{3}{p}^3)}{3^{4/3}\mathrm{\Gamma }\left(\frac{4}{3}\right)}+\frac{p^2{}_{1}F_1(\frac{2}{3};\frac{5}{3};\frac{1}{3}{p}^3)}{3^{5/3}\mathrm{\Gamma }\left(\frac{5}{3}\right)}\right),$$
$p\in ℂ$.
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_{1}F_1(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $ℂ$: complex plane, $dx$: differential of $x$, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $p$: parameter Keywords: Laplace transform Source: Gibbs (1973, Problem 72–21) Permalink: http://dlmf.nist.gov/9.10.E14 Encodings: TeX, pMML, png See also: Annotations for §9.10(v), §9.10 and Ch.9

9.10.15		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Ai}\left(-t\right)dt=\frac{1}{3}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}+\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}\right),$$
$\mathrm{\Re }p>0$,
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter Keywords: Laplace transform Source: Gibbs (1973, Problem 72–21) Permalink: http://dlmf.nist.gov/9.10.E15 Encodings: TeX, pMML, png See also: Annotations for §9.10(v), §9.10 and Ch.9

9.10.16		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-pt}\mathrm{Bi}\left(-t\right)dt=\frac{1}{\sqrt{3}}{\mathrm{e}}^{p^3/3}\left(\frac{\mathrm{\Gamma }(\frac{2}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{2}{3}\right)}-\frac{\mathrm{\Gamma }(\frac{1}{3},\frac{1}{3}{p}^3)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\right),$$
$\mathrm{\Re }p>0$.
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part and $p$: parameter Keywords: Laplace transform Source: Gibbs (1973, Problem 72–21) Permalink: http://dlmf.nist.gov/9.10.E16 Encodings: TeX, pMML, png See also: Annotations for §9.10(v), §9.10 and Ch.9

For the confluent hypergeometric function ${}_{1}F_1$ and the incomplete gamma function $\mathrm{\Gamma }$ see §§13.1, 13.2, and 8.2(i).

For Laplace transforms of products of Airy functions see Shawagfeh (1992).

9.10.17		$${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\mathrm{Ai}\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\alpha \right)}{3^{(\alpha +2)/3}\mathrm{\Gamma }\left(\frac{1}{3}\alpha +\frac{2}{3}\right)},$$
$\mathrm{\Re }\alpha >0$.
ⓘ Defines: $\alpha$: parameter (locally) Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\int$: integral and $\mathrm{\Re }$: real part Keywords: Mellin transform Source: Olver (1997b, (6.10), p. 338) Permalink: http://dlmf.nist.gov/9.10.E17 Encodings: TeX, pMML, png See also: Annotations for §9.10(vi), §9.10 and Ch.9

9.10.18		$$\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{4\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt,$$
.
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable Source: To prove this equation, combine (13.4.5) with (13.6.11) and use (5.5.3). Referenced by: (9.10.19), §9.17(iii), §9.5(ii), Erratum (V1.0.10) for Equation (9.10.18) Permalink: http://dlmf.nist.gov/9.10.E18 Encodings: TeX, pMML, png Errata (effective with 1.0.10): The original and incorrect version, $$\mathrm{Ai}\left(z\right)=\frac{z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{2^{7/2}\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt$$ taken from Schulten et al. (1979), has been corrected. Reported 2015-03-20 by Walter Gautschi See also: Annotations for §9.10(vii), §9.10 and Ch.9

9.10.19		$$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt,$$
$x>0$,
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, ${⨍}_a^b$: Cauchy principal value and $x$: real variable Source: To prove this equation, combine (9.2.10) with (9.10.18). Referenced by: §9.5(i), Erratum (V1.0.10) for Equation (9.10.19) Permalink: http://dlmf.nist.gov/9.10.E19 Encodings: TeX, pMML, png Errata (effective with 1.0.10): The original and incorrect version, $$\mathrm{Bi}\left(x\right)=\frac{x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2^{5/2}\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt$$ taken from Schulten et al. (1979), has been corrected. Reported 2015-03-20 by Walter Gautschi See also: Annotations for §9.10(vii), §9.10 and Ch.9

where the last integral is a Cauchy principal value (§1.4(v)).

9.10.20		$${\int }_0^x{\int }_0^v\mathrm{Ai}\left(t\right)dtdv=x{\int }_0^x\mathrm{Ai}\left(t\right)dt-{\mathrm{Ai}}^{\prime }\left(x\right)+{\mathrm{Ai}}^{\prime }\left(0\right),$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$, $\int$: integral, $x$: real variable and $v$: parameter Source: To verify, differentiate and refer to (9.2.1) Referenced by: 2nd item, 2nd item Permalink: http://dlmf.nist.gov/9.10.E20 Encodings: TeX, pMML, png See also: Annotations for §9.10(viii), §9.10 and Ch.9

9.10.21		$${\int }_0^x{\int }_0^v\mathrm{Bi}\left(t\right)dtdv=x{\int }_0^x\mathrm{Bi}\left(t\right)dt-{\mathrm{Bi}}^{\prime }\left(x\right)+{\mathrm{Bi}}^{\prime }\left(0\right),$$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $dx$: differential of $x$, $\int$: integral, $x$: real variable and $v$: parameter Source: To verify, differentiate and refer to (9.2.1) Referenced by: 2nd item Permalink: http://dlmf.nist.gov/9.10.E21 Encodings: TeX, pMML, png See also: Annotations for §9.10(viii), §9.10 and Ch.9

9.10.22		$${\int }_0^{\mathrm{\infty }}{\int }_t^{\mathrm{\infty }}\mathrm{\cdots }{\int }_t^{\mathrm{\infty }}\mathrm{Ai}\left(-t\right){(dt)}^n=\frac{2\cos \left(\frac{1}{3}(n-1)\pi \right)}{3^{(n+2)/3}\mathrm{\Gamma }\left(\frac{1}{3}n+\frac{2}{3}\right)},$$
$n=1,2,\mathrm{\dots }.$
ⓘ Defines: $n$: integer (locally) Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ and $\int$: integral Source: Olver (1997b, p. 344) Permalink: http://dlmf.nist.gov/9.10.E22 Encodings: TeX, pMML, png See also: Annotations for §9.10(viii), §9.10 and Ch.9

For further integrals, including the Airy transform, see §9.11(iv), Widder (1979), Prudnikov et al. (1990, §1.8.1), Prudnikov et al. (1992a, pp. 405–413), Prudnikov et al. (1992b, §4.3.25), Vallée and Soares (2010, Chapters 3, 4).