# §9.5 Integral Representations

## §9.5(i) Real Variable

 9.5.1 $\operatorname{Ai}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\cos\left(\tfrac% {1}{3}t^{3}+xt\right)\,\mathrm{d}t.$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: 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 Ch.9
 9.5.2 $\operatorname{Ai}\left(-x\right)=\frac{x^{\ifrac{1}{2}}}{\pi}\int_{-1}^{\infty% }\cos\left(x^{\ifrac{3}{2}}(\tfrac{1}{3}t^{3}+t^{2}-\tfrac{2}{3})\right)\,% \mathrm{d}t,$ $x>0$. ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: 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 Ch.9
 9.5.3 $\operatorname{Bi}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-{% \tfrac{1}{3}}t^{3}+xt\right)\,\mathrm{d}t+\frac{1}{\pi}\int_{0}^{\infty}\sin% \left(\tfrac{1}{3}t^{3}+xt\right)\,\mathrm{d}t.$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: 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 natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $(\NVar{a},\NVar{b}]$: half-closed interval, $\sin\NVar{z}$: sine function and $x$: real variable Proof sketch: 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 Ch.9

 9.5.4 $\operatorname{Ai}\left(z\right)=\frac{1}{2\pi i}\int_{\infty e^{-\pi i/3}}^{% \infty e^{\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\,\mathrm{d}t,$
 9.5.5 $\operatorname{Bi}\left(z\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty e^{\pi i/% 3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\,\mathrm{d}t+\dfrac{1}{2\pi}\int_{-% \infty}^{\infty e^{-\pi i/3}}\exp\left(\tfrac{1}{3}t^{3}-zt\right)\,\mathrm{d}t.$
 9.5.6 $\operatorname{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi}\int_{0}^{\infty}\exp% \left(-\frac{t^{3}}{3}-\frac{z^{3}}{3t^{3}}\right)\,\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{6}\pi$. ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\exp\NVar{z}$: 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: Erratum (V1.0.15) for Equation (9.5.6) 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. See also: Annotations for §9.5(ii), §9.5 and Ch.9
 9.5.7 $\operatorname{Ai}\left(z\right)=\frac{e^{-\zeta}}{\pi}\int_{0}^{\infty}\exp% \left(-z^{\ifrac{1}{2}}t^{2}\right)\cos\left(\tfrac{1}{3}t^{3}\right)\,\mathrm% {d}t,$ $|\operatorname{ph}z|<\pi$.
 9.5.8 $\operatorname{Ai}\left(z\right)=\frac{e^{-\zeta}\zeta^{\ifrac{-1}{6}}}{\sqrt{% \pi}(48)^{\ifrac{1}{6}}\Gamma\left(\frac{5}{6}\right)}\int_{0}^{\infty}e^{-t}t% ^{-\ifrac{1}{6}}\left(2+\frac{t}{\zeta}\right)^{-\ifrac{1}{6}}\,\mathrm{d}t,$ $|\operatorname{ph}z|<\frac{2}{3}\pi$.
In (9.5.7) and (9.5.8) $\zeta=\frac{2}{3}z^{\ifrac{3}{2}}$.