§9.5 Integral Representations

Permalink:: http://dlmf.nist.gov/9.5

§9.5(i) Real Variable
§9.5(ii) Complex Variable

§9.5(i) Real Variable

Notes:: See Miller (1946, p. B17) and Olver (1997b, p. 103).
Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.5.i

9.5.1 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\cos\/}\nolimits\!\left(\tfrac{1}{3}t^{3}+xt\right)dt.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral and : real variable
A&S Ref:: 10.4.32 (in different form)
Permalink:: http://dlmf.nist.gov/9.5.E1
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9.5.2 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(-x\right)=\frac{x^{{\ifrac{1}{2}}}}{\pi% }\int_{{-1}}^{\infty}\mathop{\cos\/}\nolimits\!\left(x^{{\ifrac{3}{2}}}(\tfrac% {1}{3}t^{3}+t^{2}-\tfrac{2}{3})\right)dt,$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral and : real variable
A&S Ref:: 10.4.32 (in different form)
Permalink:: http://dlmf.nist.gov/9.5.E2
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9.5.3 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\exp\/}\nolimits\!\left(-{\tfrac{1}{3}}t^{3}+xt\right)dt+\frac{1}{\pi}% \int_{0}^{\infty}\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{3}t^{3}+xt\right)dt.$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and : real variable
A&S Ref:: 10.4.33
Referenced by:: §9.12(vii)
Permalink:: http://dlmf.nist.gov/9.5.E3
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See also (9.10.19), (9.11.3), (36.9.2), and Vallée and Soares (2010, §2.1.3).

§9.5(ii) Complex Variable

Notes:: For (9.5.4) see Olver (1997b, p. 53). For (9.5.5) combine (9.2.10) and (9.5.4). For (9.5.6) see Reid (1995). For (9.5.7) see Copson (1963). (9.5.8) follows from the first of (9.6.2) and (10.32.9).
Keywords:: Airy functions
Permalink:: http://dlmf.nist.gov/9.5.ii

9.5.4 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi i}\int_{{\infty e% ^{{-\pi i/3}}}}^{{\infty e^{{\pi i/3}}}}\mathop{\exp\/}\nolimits\!\left(\tfrac% {1}{3}t^{3}-zt\right)dt,$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, : base of exponential function, $\int$ : integral and : complex variable
Referenced by:: §36.2(ii), §9.14, §9.17(iii), §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E4
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9.5.5 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi}\int_{{-\infty}}% ^{{\infty e^{{\pi i/3}}}}\mathop{\exp\/}\nolimits\!\left(\tfrac{1}{3}t^{3}-zt% \right)dt+\dfrac{1}{2\pi}\int_{{-\infty}}^{{\infty e^{{-\pi i/3}}}}\mathop{% \exp\/}\nolimits\!\left(\tfrac{1}{3}t^{3}-zt\right)dt.$

Symbols:: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, : base of exponential function, $\int$ : integral and : complex variable
Referenced by:: §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E5
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9.5.6 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{\sqrt{3}}{2\pi}\int_{0}^% {\infty}\mathop{\exp\/}\nolimits\!\left(-\frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}% \right)dt.$

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral and : complex variable
Referenced by:: §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E6
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9.5.7 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{e^{{-\zeta}}}{\pi}\int_{% 0}^{\infty}\mathop{\exp\/}\nolimits\!\left(-z^{{\ifrac{1}{2}}}t^{2}\right)% \mathop{\cos\/}\nolimits\!\left(\tfrac{1}{3}t^{3}\right)dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ .

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\mathop{\exp\/}\nolimits z$ : exponential function, : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable and $\zeta$ : change of variable
Referenced by:: §9.5(ii), §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E7
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9.5.8 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)=\frac{e^{{-\zeta}}\zeta^{{% \ifrac{-1}{6}}}}{\sqrt{\pi}(48)^{{\ifrac{1}{6}}}\mathop{\Gamma\/}\nolimits\!% \left(\frac{5}{6}\right)}\int_{0}^{\infty}e^{{-t}}t^{{-\ifrac{1}{6}}}\left(2+% \frac{t}{\zeta}\right)^{{-\ifrac{1}{6}}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{2}{3}\pi$ .

Symbols:: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ : Airy function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, : complex variable and $\zeta$ : change of variable
Referenced by:: §9.17(iii), §9.5(ii), §9.5(ii)
Permalink:: http://dlmf.nist.gov/9.5.E8
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In (9.5.7) and (9.5.8) $\zeta=\frac{2}{3}z^{{\ifrac{3}{2}}}$ .

§9.5 Integral Representations

Contents

§9.5(i) Real Variable

§9.5(ii) Complex Variable