§9.10 Integrals

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Permalink:: http://dlmf.nist.gov/9.10
See also:: Annotations for Ch.9

§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

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Keywords:: Airy functions, indefinite, integrals
Permalink:: http://dlmf.nist.gov/9.10.i
See also:: Annotations for §9.10 and Ch.9

9.10.1 $\int_{z}^{\infty}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\pi\left(% \operatorname{Ai}\left(z\right)\operatorname{Gi}'\left(z\right)-\operatorname{% Ai}'\left(z\right)\operatorname{Gi}\left(z\right)\right),$

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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 , $\int$ : integral and : 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 $\int_{-\infty}^{z}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\pi\left(% \operatorname{Ai}\left(z\right)\operatorname{Hi}'\left(z\right)-\operatorname{% Ai}'\left(z\right)\operatorname{Hi}\left(z\right)\right),$

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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 , $\int$ : integral and : 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 $\int_{-\infty}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t=\int_{0}^{z}% \operatorname{Bi}\left(t\right)\,\mathrm{d}t=\pi\left(\operatorname{Bi}'\left(% z\right)\operatorname{Gi}\left(z\right)-\operatorname{Bi}\left(z\right)% \operatorname{Gi}'\left(z\right)\right)\\ =\pi\left(\operatorname{Bi}\left(z\right)\operatorname{Hi}'\left(z\right)-% \operatorname{Bi}'\left(z\right)\operatorname{Hi}\left(z\right)\right).$

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous 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 , $\int$ : integral and : complex variable
Proof sketch:: 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 $\operatorname{Gi}$ and $\operatorname{Hi}$ see §9.12.

§9.10(ii) Asymptotic Approximations

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Keywords:: Airy functions, asymptotic approximations, integrals
Permalink:: http://dlmf.nist.gov/9.10.ii
See also:: Annotations for §9.10 and Ch.9

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

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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 , $\exp\NVar{z}$ : exponential function, $\int$ : integral and : 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 $\int_{0}^{x}\operatorname{Bi}\left(t\right)\,\mathrm{d}t\sim\pi^{-1/2}x^{-3/4}% \exp\left(\tfrac{2}{3}x^{3/2}\right),$ $x\rightarrow\infty$ .

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\sim$ : asymptotic equality, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral and : real variable
Proof sketch:: 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 $\int_{-\infty}^{x}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\pi^{-1/2}(-x)^% {-3/4}\*\cos\left(\tfrac{2}{3}(-x)^{3/2}+\tfrac{1}{4}\pi\right)+O\left(|x|^{-9% /4}\right),$ $x\rightarrow-\infty$ ,

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral and : real variable
Proof sketch:: 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 $\int_{-\infty}^{x}\operatorname{Bi}\left(t\right)\,\mathrm{d}t=\pi^{-1/2}(-x)^% {-3/4}\*\sin\left(\tfrac{2}{3}(-x)^{3/2}+\tfrac{1}{4}\pi\right)+O\left(|x|^{-9% /4}\right),$ $x\rightarrow-\infty$ .

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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 , $\int$ : integral, $\sin\NVar{z}$ : sine function and : 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).

§9.10(iii) Other Indefinite Integrals

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Permalink:: http://dlmf.nist.gov/9.10.iii
See also:: Annotations for §9.10 and Ch.9

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

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

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : complex variable and : 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 $\int z^{2}w(z)\,\mathrm{d}z=zw^{\prime}(z)-w(z),$

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Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : complex variable and : 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 $\int z^{n+3}w(z)\,\mathrm{d}z=z^{n+2}w^{\prime}(z)-(n+2)z^{n+1}w(z)+(n+1)(n+2)% \int z^{n}w(z)\,\mathrm{d}z,$ $n=0,1,2,\ldots.$

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Defines:: : index (locally)
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : complex variable and : 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

§9.10(iv) Definite Integrals

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Keywords:: Airy functions, definite, integrals
Notes:: See Olver (1997b, p. 431).
Permalink:: http://dlmf.nist.gov/9.10.iv
See also:: Annotations for §9.10 and Ch.9

9.10.11

$\int_{0}^{\infty}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\tfrac{1}{3}$ ,

$\int_{-\infty}^{0}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=\tfrac{2}{3}$ ,

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\,\mathrm{d}\NVar{x}$ : differential of 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 $\int_{-\infty}^{0}\operatorname{Bi}\left(t\right)\,\mathrm{d}t=0.$

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\,\mathrm{d}\NVar{x}$ : differential of 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

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Keywords:: Airy functions, Laplace transforms
Notes:: See Gibbs (1973, Problem 72–21), Widder (1979).
Permalink:: http://dlmf.nist.gov/9.10.v
See also:: Annotations for §9.10 and Ch.9

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and : 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 $\int_{0}^{\infty}e^{-pt}\operatorname{Ai}\left(t\right)\,\mathrm{d}t=e^{-p^{3}% /3}\left(\frac{1}{3}-\frac{p{{}_{1}F_{1}}\left(\tfrac{1}{3};\tfrac{4}{3};% \tfrac{1}{3}p^{3}\right)}{3^{4/3}\Gamma\left(\tfrac{4}{3}\right)}+\frac{p^{2}{% {}_{1}F_{1}}\left(\tfrac{2}{3};\tfrac{5}{3};\tfrac{1}{3}p^{3}\right)}{3^{5/3}% \Gamma\left(\tfrac{5}{3}\right)}\right),$ $p\in\mathbb{C}$ .

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{1}F_{1}}\left(\NVar{a};\NVar{b};\NVar{z}\right)$ : $=M\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and : 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 $\int_{0}^{\infty}e^{-pt}\operatorname{Ai}\left(-t\right)\,\mathrm{d}t={\frac{1% }{3}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{1}{3},\tfrac{1}{3}p^{3}\right)}{% \Gamma\left(\tfrac{1}{3}\right)}+\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^% {3}\right)}{\Gamma\left(\tfrac{2}{3}\right)}\right)},$ $\Re p>0$ ,

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part and : 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 $\int_{0}^{\infty}e^{-pt}\operatorname{Bi}\left(-t\right)\,\mathrm{d}t={\frac{1% }{\sqrt{3}}e^{p^{3}/3}\left(\frac{\Gamma\left(\tfrac{2}{3},\tfrac{1}{3}p^{3}% \right)}{\Gamma\left(\tfrac{2}{3}\right)}-\frac{\Gamma\left(\tfrac{1}{3},% \tfrac{1}{3}p^{3}\right)}{\Gamma\left(\tfrac{1}{3}\right)}\right)},$ $\Re p>0$ .

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part and : 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 $\Gamma$ 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

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Keywords:: Airy functions, Mellin transform
Notes:: See Olver (1997b, p. 338).
Permalink:: http://dlmf.nist.gov/9.10.vi
See also:: Annotations for §9.10 and Ch.9

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

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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 , $\int$ : 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

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Keywords:: Airy functions, Stieltjes transforms
Permalink:: http://dlmf.nist.gov/9.10.vii
See also:: Annotations for §9.10 and Ch.9

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

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase and : complex variable
Proof sketch:: 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,
$\operatorname{Ai}\left(z\right)=\frac{z^{5/4}e^{-(2/3)z^{3/2}}}{2^{7/2}\pi}\*% \int_{0}^{\infty}\frac{t^{-1/2}e^{-(2/3)t^{3/2}}\operatorname{Ai}\left(t\right% )}{z^{3/2}+t^{3/2}}\,\mathrm{d}t$
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 $\operatorname{Bi}\left(x\right)=\frac{3x^{5/4}e^{(2/3)x^{3/2}}}{2\pi}\*\pvint_% {0}^{\infty}\frac{t^{-3/4}e^{-(2/3)t^{3/2}}\operatorname{Ai}\left(t\right)}{x^% {3/2}-t^{3/2}}\,\mathrm{d}t,$

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value and : real variable
Proof sketch:: 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,
$\operatorname{Bi}\left(x\right)=\frac{x^{5/4}e^{(2/3)x^{3/2}}}{2^{5/2}\pi}\*% \pvint_{0}^{\infty}\frac{t^{-1/2}e^{-(2/3)t^{3/2}}\operatorname{Ai}\left(t% \right)}{x^{3/2}-t^{3/2}}\,\mathrm{d}t$
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(viii) Repeated Integrals

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Keywords:: Airy functions, integrals, repeated
Notes:: See Olver (1997b, pp. 342–344).
Permalink:: http://dlmf.nist.gov/9.10.viii
See also:: Annotations for §9.10 and Ch.9

9.10.20 $\int_{0}^{x}\!\!\int_{0}^{v}\operatorname{Ai}\left(t\right)\,\mathrm{d}t\,% \mathrm{d}v=x\int_{0}^{x}\operatorname{Ai}\left(t\right)\,\mathrm{d}t-% \operatorname{Ai}'\left(x\right)+\operatorname{Ai}'\left(0\right),$

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : real variable and : 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 $\int_{0}^{x}\!\!\int_{0}^{v}\operatorname{Bi}\left(t\right)\,\mathrm{d}t\,% \mathrm{d}v=x\int_{0}^{x}\operatorname{Bi}\left(t\right)\,\mathrm{d}t-% \operatorname{Bi}'\left(x\right)+\operatorname{Bi}'\left(0\right),$

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, : real variable and : 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 $\int_{0}^{\infty}\!\!\int_{t}^{\infty}\!\!\!\!\cdots\int_{t}^{\infty}% \operatorname{Ai}\left({-}t\right)(\,\mathrm{d}t)^{n}=\frac{2\cos\left(\tfrac{% 1}{3}(n-1)\pi\right)}{3^{(n+2)/3}\Gamma\left(\tfrac{1}{3}n+\tfrac{2}{3}\right)},$ $n=1,2,\ldots.$

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Defines:: : 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 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

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Keywords:: Airy transform
Permalink:: http://dlmf.nist.gov/9.10.ix
See also:: Annotations for §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).

9.9 Zeros 9.11 Products

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