# §9.10 Integrals

## §9.10(i) Indefinite Integrals

 9.10.1 $\int_{z}^{\infty}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)\mathrm{d}t=% \pi\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\mathop{\mathrm{Gi}\/}% \nolimits'\!\left(z\right)-\mathop{\mathrm{Ai}\/}\nolimits'\!\left(z\right)% \mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)\right),$
 9.10.2 $\int_{-\infty}^{z}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)\mathrm{d}t=% \pi\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\mathop{\mathrm{Hi}\/}% \nolimits'\!\left(z\right)-\mathop{\mathrm{Ai}\/}\nolimits'\!\left(z\right)% \mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)\right),$
 9.10.3 $\int_{-\infty}^{z}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)\mathrm{d}t=% \int_{0}^{z}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)\mathrm{d}t=\pi% \left(\mathop{\mathrm{Bi}\/}\nolimits'\!\left(z\right)\mathop{\mathrm{Gi}\/}% \nolimits\!\left(z\right)-\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)% \mathop{\mathrm{Gi}\/}\nolimits'\!\left(z\right)\right)\\ =\pi\left(\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)\mathop{\mathrm{Hi}\/% }\nolimits'\!\left(z\right)-\mathop{\mathrm{Bi}\/}\nolimits'\!\left(z\right)% \mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)\right).$

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

## §9.10(ii) Asymptotic Approximations

 9.10.4 $\displaystyle\int_{x}^{\infty}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)% \mathrm{d}t$ $\displaystyle\sim\tfrac{1}{2}\pi^{-1/2}x^{-3/4}\mathop{\exp\/}\nolimits\!\left% ({-}\tfrac{2}{3}x^{3/2}\right),$ $x\rightarrow\infty$, 9.10.5 $\displaystyle\int_{0}^{x}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)% \mathrm{d}t$ $\displaystyle\sim\pi^{-1/2}x^{-3/4}\mathop{\exp\/}\nolimits\!\left(\tfrac{2}{3% }x^{3/2}\right),$ $x\rightarrow\infty$.
 9.10.6 $\int_{-\infty}^{x}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)\mathrm{d}t=% \pi^{-1/2}(-x)^{-3/4}\*\mathop{\cos\/}\nolimits\!\left(\tfrac{2}{3}(-x)^{3/2}+% \tfrac{1}{4}\pi\right)+\mathop{O\/}\nolimits\!\left(|x|^{-9/4}\right),$ $x\rightarrow-\infty$,
 9.10.7 $\int_{-\infty}^{x}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)\mathrm{d}t=% \pi^{-1/2}(-x)^{-3/4}\*\mathop{\sin\/}\nolimits\!\left(\tfrac{2}{3}(-x)^{3/2}+% \tfrac{1}{4}\pi\right)+\mathop{O\/}\nolimits\!\left(|x|^{-9/4}\right),$ $x\rightarrow-\infty$.

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 Referenced by: §9.10(iii) Permalink: http://dlmf.nist.gov/9.10.E8 Encodings: TeX, pMML, png See also: Annotations for 9.10(iii) 9.10.9 $\displaystyle\int z^{2}w(z)\mathrm{d}z$ $\displaystyle=zw^{\prime}(z)-w(z),$
 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 Referenced by: §9.10(iii) Permalink: http://dlmf.nist.gov/9.10.E10 Encodings: TeX, pMML, png See also: Annotations for 9.10(iii)

## §9.10(iv) Definite Integrals

 9.10.11 $\displaystyle\int_{0}^{\infty}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)% \mathrm{d}t$ $\displaystyle=\tfrac{1}{3}$, $\displaystyle\int_{-\infty}^{0}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)% \mathrm{d}t$ $\displaystyle=\tfrac{2}{3}$, Symbols: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Referenced by: §9.10(i), §9.10(ii) Permalink: http://dlmf.nist.gov/9.10.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 9.10(iv)
 9.10.12 $\int_{-\infty}^{0}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)\mathrm{d}t=0.$ Symbols: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Referenced by: §9.10(ii) Permalink: http://dlmf.nist.gov/9.10.E12 Encodings: TeX, pMML, png See also: Annotations for 9.10(iv)

## §9.10(v) Laplace Transforms

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

For the confluent hypergeometric function $\mathop{{{}_{1}F_{1}}\/}\nolimits$ and the incomplete gamma function $\mathop{\Gamma\/}\nolimits$ 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 $\int_{0}^{\infty}t^{\alpha-1}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)% \mathrm{d}t=\frac{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)}{3^{(\alpha+% 2)/3}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}\alpha+\tfrac{2}{3}\right)},$ $\Re\alpha>0$. Defines: $\alpha$: parameter (locally) Symbols: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $\Re{}$: real part Permalink: http://dlmf.nist.gov/9.10.E17 Encodings: TeX, pMML, png See also: Annotations for 9.10(vi)

## §9.10(vii) Stieltjes Transforms

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

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

## §9.10(viii) Repeated Integrals

 9.10.20 $\int_{0}^{x}\!\!\int_{0}^{v}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)% \mathrm{d}t\mathrm{d}v=x\int_{0}^{x}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t% \right)\mathrm{d}t-\mathop{\mathrm{Ai}\/}\nolimits'\!\left(x\right)+\mathop{% \mathrm{Ai}\/}\nolimits'\!\left(0\right),$
 9.10.21 $\int_{0}^{x}\!\!\int_{0}^{v}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)% \mathrm{d}t\mathrm{d}v=x\int_{0}^{x}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t% \right)\mathrm{d}t-\mathop{\mathrm{Bi}\/}\nolimits'\!\left(x\right)+\mathop{% \mathrm{Bi}\/}\nolimits'\!\left(0\right),$
 9.10.22 $\int_{0}^{\infty}\!\!\int_{t}^{\infty}\!\!\!\!\cdots\int_{t}^{\infty}\mathop{% \mathrm{Ai}\/}\nolimits\!\left({-}t\right)(\mathrm{d}t)^{n}=\frac{2\mathop{% \cos\/}\nolimits\!\left(\tfrac{1}{3}(n-1)\pi\right)}{3^{(n+2)/3}\mathop{\Gamma% \/}\nolimits\!\left(\tfrac{1}{3}n+\tfrac{2}{3}\right)},$ $n=1,2,\ldots.$ Defines: $n$: integer (locally) Symbols: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$ and $\int$: integral Referenced by: §9.10(viii) Permalink: http://dlmf.nist.gov/9.10.E22 Encodings: TeX, pMML, png See also: Annotations for 9.10(viii)

## §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).