# §9.10 Integrals

## §9.10(i) Indefinite Integrals

 9.10.1 $\int_{z}^{\infty}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\pi\left(% \operatorname{Ai}\left(z\right)\operatorname{Gi}'\left(z\right)-\operatorname{% Ai}'\left(z\right)\operatorname{Gi}\left(z\right)\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Gi}\left(\NVar{z}\right)$: 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 Proof sketch: 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 $\int_{-\infty}^{z}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\pi\left(% \operatorname{Ai}\left(z\right)\operatorname{Hi}'\left(z\right)-\operatorname{% Ai}'\left(z\right)\operatorname{Hi}\left(z\right)\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Hi}\left(\NVar{z}\right)$: 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 Proof sketch: 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 $\int_{-\infty}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t=\int_{0}^{z}% \operatorname{Bi}\left(t\right)\,\mathrm{d}t=\pi\left(\operatorname{Bi}'\left(% z\right)\operatorname{Gi}\left(z\right)-\operatorname{Bi}\left(z\right)% \operatorname{Gi}'\left(z\right)\right)\\ =\pi\left(\operatorname{Bi}\left(z\right)\operatorname{Hi}'\left(z\right)-% \operatorname{Bi}'\left(z\right)\operatorname{Hi}\left(z\right)\right).$

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

## §9.10(ii) Asymptotic Approximations

 9.10.4 $\displaystyle\int_{x}^{\infty}\operatorname{Ai}\left(t\right)\,\mathrm{d}t$ $\displaystyle\sim\tfrac{1}{2}\pi^{-1/2}x^{-3/4}\exp\left({-}\tfrac{2}{3}x^{3/2% }\right),$ $x\rightarrow\infty$, ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\sim$: asymptotic equality, $\pi$: 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 and $x$: real variable Proof sketch: 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 $\displaystyle\int_{0}^{x}\operatorname{Bi}\left(t\right)\,\mathrm{d}t$ $\displaystyle\sim\pi^{-1/2}x^{-3/4}\exp\left(\tfrac{2}{3}x^{3/2}\right),$ $x\rightarrow\infty$.
 9.10.6 $\int_{-\infty}^{x}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\pi^{-1/2}(-x)^% {-3/4}\*\cos\left(\tfrac{2}{3}(-x)^{3/2}+\tfrac{1}{4}\pi\right)+O\left(|x|^{-9% /4}\right),$ $x\rightarrow-\infty$,
 9.10.7 $\int_{-\infty}^{x}\operatorname{Bi}\left(t\right)\,\mathrm{d}t=\pi^{-1/2}(-x)^% {-3/4}\*\sin\left(\tfrac{2}{3}(-x)^{3/2}+\tfrac{1}{4}\pi\right)+O\left(|x|^{-9% /4}\right),$ $x\rightarrow-\infty$. ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: 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 Proof sketch: 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).

## §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 Proof sketch: 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 $\displaystyle\int z^{2}w(z)\,\mathrm{d}z$ $\displaystyle=zw^{\prime}(z)-w(z),$ ⓘ Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $z$: complex variable and $w$: ODE solution Proof sketch: 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 $\int z^{n+3}w(z)\,\mathrm{d}z=z^{n+2}w^{\prime}(z)-(n+2)z^{n+1}w(z)+(n+1)(n+2)% \int z^{n}w(z)\,\mathrm{d}z,$ $n=0,1,2,\ldots.$ ⓘ Defines: $n$: index (locally) Symbols: $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $z$: complex variable and $w$: ODE solution Proof sketch: 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(iv) Definite Integrals

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

## §9.10(v) Laplace Transforms

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

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

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

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

## §9.10(viii) Repeated Integrals

 9.10.20 $\int_{0}^{x}\!\!\int_{0}^{v}\operatorname{Ai}\left(t\right)\,\mathrm{d}t\,% \mathrm{d}v=x\int_{0}^{x}\operatorname{Ai}\left(t\right)\,\mathrm{d}t-% \operatorname{Ai}'\left(x\right)+\operatorname{Ai}'\left(0\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $x$: real variable and $v$: parameter Proof sketch: 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 $\int_{0}^{x}\!\!\int_{0}^{v}\operatorname{Bi}\left(t\right)\,\mathrm{d}t\,% \mathrm{d}v=x\int_{0}^{x}\operatorname{Bi}\left(t\right)\,\mathrm{d}t-% \operatorname{Bi}'\left(x\right)+\operatorname{Bi}'\left(0\right),$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $x$: real variable and $v$: parameter Proof sketch: 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 $\int_{0}^{\infty}\!\!\int_{t}^{\infty}\!\!\!\!\cdots\int_{t}^{\infty}% \operatorname{Ai}\left({-}t\right)(\,\mathrm{d}t)^{n}=\frac{2\cos\left(\tfrac{% 1}{3}(n-1)\pi\right)}{3^{(n+2)/3}\Gamma\left(\tfrac{1}{3}n+\tfrac{2}{3}\right)},$ $n=1,2,\ldots.$ ⓘ Defines: $n$: integer (locally) Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\Gamma\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: 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 Ch.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).