# §9.10 Integrals

## §9.10(i) Indefinite Integrals

 9.10.1 $\int_{z}^{\infty}\mathrm{Ai}\left(t\right)\mathrm{d}t=\pi\left(\mathrm{Ai}% \left(z\right)\mathrm{Gi}'\left(z\right)-\mathrm{Ai}'\left(z\right)\mathrm{Gi}% \left(z\right)\right),$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Gi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: 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 9
 9.10.2 $\int_{-\infty}^{z}\mathrm{Ai}\left(t\right)\mathrm{d}t=\pi\left(\mathrm{Ai}% \left(z\right)\mathrm{Hi}'\left(z\right)-\mathrm{Ai}'\left(z\right)\mathrm{Hi}% \left(z\right)\right),$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{Hi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: 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 9
 9.10.3 $\int_{-\infty}^{z}\mathrm{Bi}\left(t\right)\mathrm{d}t=\int_{0}^{z}\mathrm{Bi}% \left(t\right)\mathrm{d}t=\pi\left(\mathrm{Bi}'\left(z\right)\mathrm{Gi}\left(% z\right)-\mathrm{Bi}\left(z\right)\mathrm{Gi}'\left(z\right)\right)\\ =\pi\left(\mathrm{Bi}\left(z\right)\mathrm{Hi}'\left(z\right)-\mathrm{Bi}'% \left(z\right)\mathrm{Hi}\left(z\right)\right).$

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

## §9.10(ii) Asymptotic Approximations

 9.10.4 $\displaystyle\int_{x}^{\infty}\mathrm{Ai}\left(t\right)\mathrm{d}t$ $\displaystyle\sim\tfrac{1}{2}\pi^{-1/2}x^{-3/4}\exp\left({-}\tfrac{2}{3}x^{3/2% }\right),$ $x\rightarrow\infty$, 9.10.5 $\displaystyle\int_{0}^{x}\mathrm{Bi}\left(t\right)\mathrm{d}t$ $\displaystyle\sim\pi^{-1/2}x^{-3/4}\exp\left(\tfrac{2}{3}x^{3/2}\right),$ $x\rightarrow\infty$.
 9.10.6 $\int_{-\infty}^{x}\mathrm{Ai}\left(t\right)\mathrm{d}t=\pi^{-1/2}(-x)^{-3/4}\*% \cos\left(\tfrac{2}{3}(-x)^{3/2}+\tfrac{1}{4}\pi\right)+O\left(|x|^{-9/4}% \right),$ $x\rightarrow-\infty$,
 9.10.7 $\int_{-\infty}^{x}\mathrm{Bi}\left(t\right)\mathrm{d}t=\pi^{-1/2}(-x)^{-3/4}\*% \sin\left(\tfrac{2}{3}(-x)^{3/2}+\tfrac{1}{4}\pi\right)+O\left(|x|^{-9/4}% \right),$ $x\rightarrow-\infty$. ⓘ Symbols: $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $O\left(\NVar{x}\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: 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 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).

## §9.10(iii) Other Indefinite Integrals

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

 9.10.8 $\displaystyle\int zw(z)\mathrm{d}z$ $\displaystyle=w^{\prime}(z),$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: 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 9 9.10.9 $\displaystyle\int z^{2}w(z)\mathrm{d}z$ $\displaystyle=zw^{\prime}(z)-w(z),$ ⓘ Symbols: $\mathrm{d}\NVar{x}$: 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 9
 9.10.10 $\int z^{n+3}w(z)\mathrm{d}z=z^{n+2}w^{\prime}(z)-(n+2)z^{n+1}w(z)+(n+1)(n+2)% \int z^{n}w(z)\mathrm{d}z,$ $n=0,1,2,\ldots.$ ⓘ Defines: $n$: index (locally) Symbols: $\mathrm{d}\NVar{x}$: 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 9

## §9.10(iv) Definite Integrals

 9.10.11 $\displaystyle\int_{0}^{\infty}\mathrm{Ai}\left(t\right)\mathrm{d}t$ $\displaystyle=\tfrac{1}{3}$, $\displaystyle\int_{-\infty}^{0}\mathrm{Ai}\left(t\right)\mathrm{d}t$ $\displaystyle=\tfrac{2}{3}$, ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{d}\NVar{x}$: 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 9
 9.10.12 $\int_{-\infty}^{0}\mathrm{Bi}\left(t\right)\mathrm{d}t=0.$ ⓘ Symbols: $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $\mathrm{d}\NVar{x}$: 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 9

## §9.10(v) Laplace Transforms

 9.10.13 $\int_{-\infty}^{\infty}e^{pt}\mathrm{Ai}\left(t\right)\mathrm{d}t=e^{p^{3}/3},$ $\Re p>0$.
 9.10.14 $\int_{0}^{\infty}e^{-pt}\mathrm{Ai}\left(t\right)\mathrm{d}t=e^{-p^{3}/3}\left% (\frac{1}{3}-\frac{p{{}_{1}F_{1}}\left(\tfrac{1}{3};\tfrac{4}{3};\tfrac{1}{3}p% ^{3}\right)}{3^{4/3}\Gamma\left(\tfrac{4}{3}\right)}+\frac{p^{2}{{}_{1}F_{1}}% \left(\tfrac{2}{3};\tfrac{5}{3};\tfrac{1}{3}p^{3}\right)}{3^{5/3}\Gamma\left(% \tfrac{5}{3}\right)}\right),$ $p\in\mathbb{C}$.
 9.10.15 $\int_{0}^{\infty}e^{-pt}\mathrm{Ai}\left(-t\right)\mathrm{d}t={\frac{1}{3}e^{p% ^{3}/3}\left(\frac{\Gamma\left(\tfrac{1}{3},\tfrac{1}{3}p^{3}\right)}{\Gamma% \left(\tfrac{1}{3}\right)}+\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^{3}% \right)}{\Gamma\left(\tfrac{2}{3}\right)}\right)},$ $\Re p>0$,
 9.10.16 $\int_{0}^{\infty}e^{-pt}\mathrm{Bi}\left(-t\right)\mathrm{d}t={\frac{1}{\sqrt{% 3}}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^{3}\right)}{% \Gamma\left(\tfrac{2}{3}\right)}-\frac{\Gamma\left(\tfrac{1}{3},\tfrac{1}{3}p^% {3}\right)}{\Gamma\left(\tfrac{1}{3}\right)}\right)},$ $\Re p>0$.

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

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

## §9.10(vi) Mellin Transform

 9.10.17 $\int_{0}^{\infty}t^{\alpha-1}\mathrm{Ai}\left(t\right)\mathrm{d}t=\frac{\Gamma% \left(\alpha\right)}{3^{(\alpha+2)/3}\Gamma\left(\tfrac{1}{3}\alpha+\tfrac{2}{% 3}\right)},$ $\Re\alpha>0$. ⓘ Defines: $\alpha$: parameter (locally) Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\Gamma\left(\NVar{z}\right)$: gamma function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\Re$: real part 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 9

## §9.10(vii) Stieltjes Transforms

9.10.18 $\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}e^{-(2/3)z^{3/2}}}{4\pi}\*\int_{0}^{% \infty}\frac{t^{-3/4}e^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/% 2}}\mathrm{d}t,$
$|\operatorname{ph}z|<\tfrac{2}{3}\pi$.
9.10.19 $\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}e^{(2/3)x^{3/2}}}{2\pi}\*\pvint_{0}^{% \infty}\frac{t^{-3/4}e^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/% 2}}\mathrm{d}t,$
$x>0$,

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

## §9.10(viii) Repeated Integrals

 9.10.20 $\int_{0}^{x}\!\!\int_{0}^{v}\mathrm{Ai}\left(t\right)\mathrm{d}t\mathrm{d}v=x% \int_{0}^{x}\mathrm{Ai}\left(t\right)\mathrm{d}t-\mathrm{Ai}'\left(x\right)+% \mathrm{Ai}'\left(0\right),$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $x$: real variable and $v$: parameter Source: To verify, differentiate and refer to (9.2.1) Referenced by: §9.18(v), §9.19(ii) Permalink: http://dlmf.nist.gov/9.10.E20 Encodings: TeX, pMML, png See also: Annotations for 9.10(viii), 9.10 and 9
 9.10.21 $\int_{0}^{x}\!\!\int_{0}^{v}\mathrm{Bi}\left(t\right)\mathrm{d}t\mathrm{d}v=x% \int_{0}^{x}\mathrm{Bi}\left(t\right)\mathrm{d}t-\mathrm{Bi}'\left(x\right)+% \mathrm{Bi}'\left(0\right),$ ⓘ Symbols: $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $x$: real variable and $v$: parameter Source: To verify, differentiate and refer to (9.2.1) Referenced by: §9.19(ii) Permalink: http://dlmf.nist.gov/9.10.E21 Encodings: TeX, pMML, png See also: Annotations for 9.10(viii), 9.10 and 9
 9.10.22 $\int_{0}^{\infty}\!\!\int_{t}^{\infty}\!\!\!\!\cdots\int_{t}^{\infty}\mathrm{% Ai}\left({-}t\right)(\mathrm{d}t)^{n}=\frac{2\cos\left(\tfrac{1}{3}(n-1)\pi% \right)}{3^{(n+2)/3}\Gamma\left(\tfrac{1}{3}n+\tfrac{2}{3}\right)},$ $n=1,2,\ldots.$ ⓘ Defines: $n$: integer (locally) Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: 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 9

## §9.10(ix) Compendia

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).