# §9.5 Integral Representations

## §9.5(i) Real Variable

 9.5.1 $\mathrm{Ai}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\cos\left(\tfrac{1}{3}% t^{3}+xt\right)\mathrm{d}t.$ ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $x$: real variable Source: Olver (1997b, p. 53) A&S Ref: 10.4.32 (in different form) Permalink: http://dlmf.nist.gov/9.5.E1 Encodings: TeX, pMML, png See also: Annotations for 9.5(i), 9.5 and 9
 9.5.2 $\mathrm{Ai}\left(-x\right)=\frac{x^{\ifrac{1}{2}}}{\pi}\int_{-1}^{\infty}\cos% \left(x^{\ifrac{3}{2}}(\tfrac{1}{3}t^{3}+t^{2}-\tfrac{2}{3})\right)\mathrm{d}t,$ $x>0$. ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $x$: real variable Source: Olver (1997b, p. 103) A&S Ref: 10.4.32 (in different form) Permalink: http://dlmf.nist.gov/9.5.E2 Encodings: TeX, pMML, png See also: Annotations for 9.5(i), 9.5 and 9
 9.5.3 $\mathrm{Bi}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-{\tfrac{1}{% 3}}t^{3}+xt\right)\mathrm{d}t+\frac{1}{\pi}\int_{0}^{\infty}\sin\left(\tfrac{1% }{3}t^{3}+xt\right)\mathrm{d}t.$ ⓘ Symbols: $\mathrm{Bi}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b})$: half-closed interval, $\mathrm{d}\NVar{x}$: differential of $x$, $\exp\NVar{z}$: exponential function, $\mathrm{e}$: base of exponential function, $\int$: integral, $(\NVar{a},\NVar{b}]$: half-closed interval, $\sin\NVar{z}$: sine function and $x$: real variable Source: Use (9.5.5) with the substitions: for the paths $(-\infty,0]$ use $t=-\tau$, for the paths $[0,\infty{\mathrm{e}^{\pm\pi\mathrm{i}/3}})$ use $t={\mathrm{e}^{\pm\pi\mathrm{i}/3}}\tau$. For the resulting integrals use (4.14.1) and (4.14.2). A&S Ref: 10.4.33 Referenced by: 9.12.19 Permalink: http://dlmf.nist.gov/9.5.E3 Encodings: TeX, pMML, png See also: Annotations for 9.5(i), 9.5 and 9

See also (9.10.19), (9.11.3), (36.9.2), and Vallée and Soares (2010, §2.1.3).

## §9.5(ii) Complex Variable

 9.5.4 $\mathrm{Ai}\left(z\right)=\frac{1}{2\pi i}\int_{\infty e^{-\pi i/3}}^{\infty e% ^{\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\mathrm{d}t,$
 9.5.5 $\mathrm{Bi}\left(z\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty e^{\pi i/3}}% \exp\left(\tfrac{1}{3}t^{3}-zt\right)\mathrm{d}t+\dfrac{1}{2\pi}\int_{-\infty}% ^{\infty e^{-\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\mathrm{d}t.$
 9.5.6 $\mathrm{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi}\int_{0}^{\infty}\exp\left(-% \frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}\right)\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{6}\pi$. ⓘ Symbols: $\mathrm{Ai}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\exp\NVar{z}$: exponential function, $\int$: integral, $\operatorname{ph}$: phase and $z$: complex variable Source: Reid (1995, (5.4), p. 170, with change of variable and using analytic continuation to complex $z$) Referenced by: Changes Permalink: http://dlmf.nist.gov/9.5.E6 Encodings: TeX, pMML, png Clarification (effective with 1.0.15): The validity constraint $|\operatorname{ph}z|<\tfrac{1}{6}\pi$ was added. Reported 2017-03-24 See also: Annotations for 9.5(ii), 9.5 and 9
 9.5.7 $\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}}{\pi}\int_{0}^{\infty}\exp\left(-z^% {\ifrac{1}{2}}t^{2}\right)\cos\left(\tfrac{1}{3}t^{3}\right)\mathrm{d}t,$ $|\operatorname{ph}z|<\pi$.
 9.5.8 $\mathrm{Ai}\left(z\right)=\frac{e^{-\zeta}\zeta^{\ifrac{-1}{6}}}{\sqrt{\pi}(48% )^{\ifrac{1}{6}}\Gamma\left(\frac{5}{6}\right)}\int_{0}^{\infty}e^{-t}t^{-% \ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-\ifrac{1}{6}}\mathrm{d}t,$ $|\operatorname{ph}z|<\frac{2}{3}\pi$.

In (9.5.7) and (9.5.8) $\zeta=\frac{2}{3}z^{\ifrac{3}{2}}$.