§9.10 Integrals

Permalink:: http://dlmf.nist.gov/9.10

§9.10(i) Indefinite Integrals
§9.10(ii) Asymptotic Approximations
§9.10(iii) Other Indefinite Integrals
§9.10(iv) Definite Integrals
§9.10(v) Laplace Transforms
§9.10(vi) Mellin Transform
§9.10(vii) Stieltjes Transforms
§9.10(viii) Repeated Integrals
§9.10(ix) Compendia

§9.10(i) Indefinite Integrals

Notes:: For (9.10.1) combine (9.12.4) and its differentiated form with the first of (9.10.11). For (9.10.2) combine (9.12.11) and its differentiated form with (9.10.1), then apply (9.2.7) and (9.10.11). (9.10.3) is proved in a similar manner.
Keywords:: Airy functions, integrals
Permalink:: http://dlmf.nist.gov/9.10.i

9.10.1 $\int_{z}^{{\infty}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)dt=\pi\left(% \mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right){\mathop{\mathrm{Gi}\/}% \nolimits^{{\prime}}}\!\left(z\right)-{\mathop{\mathrm{Ai}\/}\nolimits^{{% \prime}}}\!\left(z\right)\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)$ : Scorer function (inhomogeneous Airy function), : differential of , $\int$ : integral and : complex variable
Referenced by:: §9.10(i)
Permalink:: http://dlmf.nist.gov/9.10.E1
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9.10.2 $\int_{{-\infty}}^{z}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)dt=\pi\left% (\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right){\mathop{\mathrm{Hi}\/}% \nolimits^{{\prime}}}\!\left(z\right)-{\mathop{\mathrm{Ai}\/}\nolimits^{{% \prime}}}\!\left(z\right)\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$ : Scorer function (inhomogeneous Airy function), : differential of , $\int$ : integral and : complex variable
Referenced by:: §9.10(i)
Permalink:: http://dlmf.nist.gov/9.10.E2
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9.10.3 $\int_{{-\infty}}^{z}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)dt=\int_{0}% ^{z}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)dt=\pi\left({\mathop{% \mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(z\right)\mathop{\mathrm{Gi}\/}% \nolimits\!\left(z\right)-\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right){% \mathop{\mathrm{Gi}\/}\nolimits^{{\prime}}}\!\left(z\right)\right)\\ =\pi\left(\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right){\mathop{\mathrm{Hi}% \/}\nolimits^{{\prime}}}\!\left(z\right)-{\mathop{\mathrm{Bi}\/}\nolimits^{{% \prime}}}\!\left(z\right)\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)\right).$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)$ : Scorer function (inhomogeneous Airy function), $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$ : Scorer function (inhomogeneous Airy function), : differential of , $\int$ : integral and : complex variable
Referenced by:: §9.10(i)
Permalink:: http://dlmf.nist.gov/9.10.E3
Encodings:: TeX, pMML, png

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

§9.10(ii) Asymptotic Approximations

Notes:: Integrate the leading terms of the asymptotic expansions given in §9.7(ii) and use (9.10.11), (9.10.12).
Keywords:: Airy functions, integrals
Permalink:: http://dlmf.nist.gov/9.10.ii

9.10.4 $\int_{x}^{{\infty}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)dt\sim\tfrac% {1}{2}\pi^{{-1/2}}x^{{-3/4}}\mathop{\exp\/}\nolimits\!\left({-}\tfrac{2}{3}x^{% {3/2}}\right),$ $x\rightarrow\infty$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\sim$ : asymptotic equality and : real variable
Referenced by:: §9.10(ii)
Permalink:: http://dlmf.nist.gov/9.10.E4
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9.10.5 $\int_{0}^{x}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)dt\sim\pi^{{-1/2}}x% ^{{-3/4}}\mathop{\exp\/}\nolimits\!\left(\tfrac{2}{3}x^{{3/2}}\right),$ $x\rightarrow\infty$ .

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\sim$ : asymptotic equality and : real variable
Permalink:: http://dlmf.nist.gov/9.10.E5
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9.10.6 $\int_{{-\infty}}^{x}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)dt=\pi^{{-1% /2}}(-x)^{{-3/4}}\*\mathop{\cos\/}\nolimits\!\left(\tfrac{2}{3}(-x)^{{3/2}}+% \tfrac{1}{4}\pi\right)+\mathop{O\/}\nolimits\!\left(|x|^{{-9/4}}\right),$ $x\rightarrow-\infty$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral and : real variable
Permalink:: http://dlmf.nist.gov/9.10.E6
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9.10.7 $\int_{{-\infty}}^{x}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)dt=\pi^{{-1% /2}}(-x)^{{-3/4}}\*\mathop{\sin\/}\nolimits\!\left(\tfrac{2}{3}(-x)^{{3/2}}+% \tfrac{1}{4}\pi\right)+\mathop{O\/}\nolimits\!\left(|x|^{{-9/4}}\right),$ $x\rightarrow-\infty$ .

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and : real variable
Referenced by:: §9.10(ii)
Permalink:: http://dlmf.nist.gov/9.10.E7
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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

Notes:: To verify (9.10.8)–(9.10.10) differentiate and refer to (9.2.1).
Permalink:: http://dlmf.nist.gov/9.10.iii

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

9.10.8 $\int zw(z)dz=w^{{\prime}}(z),$

Symbols:: : differential of , $\int$ : integral, : complex variable and : ODE solution
Referenced by:: §9.10(iii)
Permalink:: http://dlmf.nist.gov/9.10.E8
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9.10.9 $\int z^{2}w(z)dz=zw^{{\prime}}(z)-w(z),$

Symbols:: : differential of , $\int$ : integral, : complex variable and : ODE solution
Permalink:: http://dlmf.nist.gov/9.10.E9
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9.10.10 $\int z^{{n+3}}w(z)dz=z^{{n+2}}w^{{\prime}}(z)-(n+2)z^{{n+1}}w(z)+(n+1)(n+2)% \int z^{n}w(z)dz,$ $n=0,1,2,\ldots.$

Symbols:: : differential of , $\int$ : integral, : complex variable, : index and : ODE solution
Referenced by:: §9.10(iii)
Permalink:: http://dlmf.nist.gov/9.10.E10
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§9.10(iv) Definite Integrals

Notes:: See Olver (1997b, p. 431).
Keywords:: Airy functions, integrals
Permalink:: http://dlmf.nist.gov/9.10.iv

9.10.11

$\int_{0}^{\infty}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)dt=\tfrac{1}{3}$ ,

$\int_{{-\infty}}^{0}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)dt=\tfrac{2% }{3}$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of and $\int$ : integral
Referenced by:: §9.10(i), §9.10(ii)
Permalink:: http://dlmf.nist.gov/9.10.E11
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9.10.12 $\int_{{-\infty}}^{0}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)dt=0.$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of and $\int$ : integral
Referenced by:: §9.10(ii)
Permalink:: http://dlmf.nist.gov/9.10.E12
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§9.10(v) Laplace Transforms

Notes:: See Gibbs (1973, Problem 72–21). For (9.10.13) see Widder (1979).
Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.10.v

9.10.13 $\int_{{-\infty}}^{{\infty}}e^{{pt}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t% \right)dt=e^{{p^{3}/3}},$ $\Re p>0$ .

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and : parameter
Referenced by:: §9.10(v)
Permalink:: http://dlmf.nist.gov/9.10.E13
Encodings:: TeX, pMML, png

9.10.14 $\int_{0}^{{\infty}}e^{{-pt}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)dt=% e^{{-p^{3}/3}}\left(\frac{1}{3}-\frac{p\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits\!% \left(\tfrac{1}{3};\tfrac{4}{3};\tfrac{1}{3}p^{3}\right)}{3^{{4/3}}\mathop{% \Gamma\/}\nolimits\!\left(\tfrac{4}{3}\right)}+\frac{p^{2}\mathop{{{}_{{1}}F_{% {1}}}\/}\nolimits\!\left(\tfrac{2}{3};\tfrac{5}{3};\tfrac{1}{3}p^{3}\right)}{3% ^{{5/3}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{5}{3}\right)}\right),$ $p\in\Complex$ .

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},% \dots,b_{q}};z\right)$ : generalized hypergeometric function, $\Complex$ : complex plane (excluding infinity), : differential of , $\in$ : element of, : base of exponential function, $\int$ : integral and : parameter
Permalink:: http://dlmf.nist.gov/9.10.E14
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9.10.15 $\int_{0}^{{\infty}}e^{{-pt}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(-t\right)dt% ={\frac{1}{3}e^{{p^{3}/3}}\left(\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac% {1}{3},\tfrac{1}{3}p^{3}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3% }\right)}+\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3},\tfrac{1}{3}p^{% 3}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}\right)}\right)},$ $\Re p>0$ ,

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part and : parameter
Permalink:: http://dlmf.nist.gov/9.10.E15
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9.10.16 $\int_{0}^{{\infty}}e^{{-pt}}\mathop{\mathrm{Bi}\/}\nolimits\!\left(-t\right)dt% ={\frac{1}{\sqrt{3}}e^{{p^{3}/3}}\left(\frac{\mathop{\Gamma\/}\nolimits\!\left% (\tfrac{2}{3},\tfrac{1}{3}p^{3}\right)}{\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{2}{3}\right)}-\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3},% \tfrac{1}{3}p^{3}\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}\right% )}\right)},$ $\Re p>0$ .

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\mathop{\Gamma\/}\nolimits\!\left(a,z\right)$ : incomplete gamma function, $\int$ : integral, $\realpart{}$ : real part and : parameter
Permalink:: http://dlmf.nist.gov/9.10.E16
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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

Notes:: See Olver (1997b, p. 338).
Keywords:: Airy functions, Mellin transform
Permalink:: http://dlmf.nist.gov/9.10.vi

9.10.17 $\int_{0}^{{\infty}}t^{{\alpha-1}}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t% \right)dt=\frac{\mathop{\Gamma\/}\nolimits\!\left(\alpha\right)}{3^{{(\alpha+2% )/3}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}\alpha+\tfrac{2}{3}\right)},$ $\Re\alpha>0$ .

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , $\int$ : integral, $\realpart{}$ : real part and $\alpha$ : parameter
Permalink:: http://dlmf.nist.gov/9.10.E17
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§9.10(vii) Stieltjes Transforms

Notes:: See Schulten et al. (1979).
Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.10.vii

9.10.18 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{z^{{5/4}}e^{{-(2/3)z^{{3% /2}}}}}{2^{{7/2}}\pi}\*\int_{0}^{{\infty}}\frac{t^{{-1/2}}e^{{-(2/3)t^{{3/2}}}% }\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)}{z^{{3/2}}+t^{{3/2}}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{2}{3}\pi$ .

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and : complex variable
Referenced by:: §9.17(iii), §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.10.E18
Encodings:: TeX, pMML, png

9.10.19 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)=\frac{x^{{5/4}}e^{{(2/3)x^{{3/% 2}}}}}{2^{{5/2}}\pi}\*\pvint_{0}^{{\infty}}\frac{t^{{-1/2}}e^{{-(2/3)t^{{3/2}}% }}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)}{x^{{3/2}}-t^{{3/2}}}dt,$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , : base of exponential function, $\pvint_{a}^{b}$ : Cauchy principal value and : real variable
Referenced by:: §9.5(i)
Permalink:: http://dlmf.nist.gov/9.10.E19
Encodings:: TeX, pMML, png

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

§9.10(viii) Repeated Integrals

Notes:: To verify (9.10.20) and (9.10.21) differentiate, and refer to (9.2.1). For (9.10.22) see Olver (1997b, pp. 342–344).
Keywords:: Airy functions, integrals
Permalink:: http://dlmf.nist.gov/9.10.viii

9.10.20 $\int_{0}^{x}\!\!\int_{0}^{v}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)% dtdv=x\int_{0}^{x}\mathop{\mathrm{Ai}\/}\nolimits\!\left(t\right)dt-{\mathop{% \mathrm{Ai}\/}\nolimits^{{\prime}}}\!\left(x\right)+{\mathop{\mathrm{Ai}\/}% \nolimits^{{\prime}}}\!\left(0\right),$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , $\int$ : integral, : real variable and : parameter
Referenced by:: §9.10(viii), §9.18(v), §9.19(ii)
Permalink:: http://dlmf.nist.gov/9.10.E20
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9.10.21 $\int_{0}^{x}\!\!\int_{0}^{v}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)% dtdv=x\int_{0}^{x}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)dt-{\mathop{% \mathrm{Bi}\/}\nolimits^{{\prime}}}\!\left(x\right)+{\mathop{\mathrm{Bi}\/}% \nolimits^{{\prime}}}\!\left(0\right),$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , $\int$ : integral, : real variable and : parameter
Referenced by:: §9.10(viii), §9.19(ii)
Permalink:: http://dlmf.nist.gov/9.10.E21
Encodings:: TeX, pMML, png

9.10.22 $\int_{0}^{{\infty}}\!\!\int_{t}^{{\infty}}\!\!\!\!\cdots\int_{t}^{{\infty}}% \mathop{\mathrm{Ai}\/}\nolimits\!\left({-}t\right)(dt)^{n}=\frac{2\mathop{\cos% \/}\nolimits\!\left(\tfrac{1}{3}(n-1)\pi\right)}{3^{{(n+2)/3}}\mathop{\Gamma\/% }\nolimits\!\left(\tfrac{1}{3}n+\tfrac{2}{3}\right)},$ $n=1,2,\ldots.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral and : integer
Referenced by:: §9.10(viii)
Permalink:: http://dlmf.nist.gov/9.10.E22
Encodings:: TeX, pMML, png

§9.10(ix) Compendia

Keywords:: Airy transform
Permalink:: http://dlmf.nist.gov/9.10.ix

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).