§9.6 Relations to Other Functions

§9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions

For the notation see §§10.2(ii) and 10.25(ii). With

 9.6.1 $\zeta=\tfrac{2}{3}z^{3/2},$ Defines: $\zeta(z)$: change of variable (locally) Symbols: $z$: complex variable Referenced by: §9.6(iii), §9.6(iii), §9.7(iv) Permalink: http://dlmf.nist.gov/9.6.E1 Encodings: TeX, pMML, png See also: Annotations for 9.6(i)
 9.6.2 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ $\displaystyle=\pi^{-1}\sqrt{z/3}\mathop{K_{\pm 1/3}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=\tfrac{1}{3}\sqrt{z}\left(\mathop{I_{-1/3}\/}\nolimits\!\left(% \zeta\right)-\mathop{I_{1/3}\/}\nolimits\!\left(\zeta\right)\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{z/3}e^{2\pi i/3}\mathop{{H^{(1)}_{1/3}}\/}% \nolimits\!\left(\zeta e^{\pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{z/3}e^{\pi i/3}\mathop{{H^{(1)}_{-1/3}}\/}% \nolimits\!\left(\zeta e^{\pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{z/3}e^{-2\pi i/3}\mathop{{H^{(2)}_{1/3}}\/}% \nolimits\!\left(\zeta e^{-\pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{z/3}e^{-\pi i/3}\mathop{{H^{(2)}_{-1/3}}\/}% \nolimits\!\left(\zeta e^{-\pi i/2}\right),$ 9.6.3 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits'\!\left(z\right)$ $\displaystyle=-\pi^{-1}(z/\sqrt{3})\mathop{K_{\pm 2/3}\/}\nolimits\!\left(% \zeta\right)$ $\displaystyle=(z/3)\left(\mathop{I_{2/3}\/}\nolimits\!\left(\zeta\right)-% \mathop{I_{-2/3}\/}\nolimits\!\left(\zeta\right)\right)$ $\displaystyle=\tfrac{1}{2}(z/\sqrt{3})e^{-\pi i/6}\mathop{{H^{(1)}_{2/3}}\/}% \nolimits\!\left(\zeta e^{\pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}(z/\sqrt{3})e^{-5\pi i/6}\mathop{{H^{(1)}_{-2/3}}\/}% \nolimits\!\left(\zeta e^{\pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}(z/\sqrt{3})e^{\pi i/6}\mathop{{H^{(2)}_{2/3}}\/}% \nolimits\!\left(\zeta e^{-\pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}(z/\sqrt{3})e^{5\pi i/6}\mathop{{H^{(2)}_{-2/3}}\/}% \nolimits\!\left(\zeta e^{-\pi i/2}\right),$ 9.6.4 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ $\displaystyle=\sqrt{z/3}\left(\mathop{I_{1/3}\/}\nolimits\!\left(\zeta\right)+% \mathop{I_{-1/3}\/}\nolimits\!\left(\zeta\right)\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/6}\mathop{{H^{(1)}_{1/3}}\/% }\nolimits\!\left(\zeta e^{-\pi i/2}\right)+e^{-\pi i/6}\mathop{{H^{(2)}_{1/3}% }\/}\nolimits\!\left(\zeta e^{\pi i/2}\right)\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{z/3}\left(e^{-\pi i/6}\mathop{{H^{(1)}_{-1/3}}% \/}\nolimits\!\left(\zeta e^{-\pi i/2}\right)+e^{\pi i/6}\mathop{{H^{(2)}_{-1/% 3}}\/}\nolimits\!\left(\zeta e^{\pi i/2}\right)\right),$ 9.6.5 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits'\!\left(z\right)$ $\displaystyle=(z/\sqrt{3})\left(\mathop{I_{2/3}\/}\nolimits\!\left(\zeta\right% )+\mathop{I_{-2/3}\/}\nolimits\!\left(\zeta\right)\right)$ $\displaystyle=\tfrac{1}{2}(z/\sqrt{3})\left(e^{\pi i/3}\mathop{{H^{(1)}_{2/3}}% \/}\nolimits\!\left(\zeta e^{-\pi i/2}\right)+e^{-\pi i/3}\mathop{{H^{(2)}_{2/% 3}}\/}\nolimits\!\left(\zeta e^{\pi i/2}\right)\right)$ $\displaystyle=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/3}\mathop{{H^{(1)}_{-2/3% }}\/}\nolimits\!\left(\zeta e^{-\pi i/2}\right)+e^{\pi i/3}\mathop{{H^{(2)}_{-% 2/3}}\/}\nolimits\!\left(\zeta e^{\pi i/2}\right)\right),$
 9.6.6 $\mathop{\mathrm{Ai}\/}\nolimits\!\left(-z\right)=(\sqrt{z}/3)\left(\mathop{J_{% 1/3}\/}\nolimits\!\left(\zeta\right)+\mathop{J_{-1/3}\/}\nolimits\!\left(\zeta% \right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/6}\mathop{{H^{(1)}_{1/3}}% \/}\nolimits\!\left(\zeta\right)+e^{-\pi i/6}\mathop{{H^{(2)}_{1/3}}\/}% \nolimits\!\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{-\pi i/6}% \mathop{{H^{(1)}_{-1/3}}\/}\nolimits\!\left(\zeta\right)+e^{\pi i/6}\mathop{{H% ^{(2)}_{-1/3}}\/}\nolimits\!\left(\zeta\right)\right),$
 9.6.7 $\mathop{\mathrm{Ai}\/}\nolimits'\!\left(-z\right)=(z/3)\left(\mathop{J_{2/3}\/% }\nolimits\!\left(\zeta\right)-\mathop{J_{-2/3}\/}\nolimits\!\left(\zeta\right% )\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/6}\mathop{{H^{(1)}_{2/3}}\/}% \nolimits\!\left(\zeta\right)+e^{\pi i/6}\mathop{{H^{(2)}_{2/3}}\/}\nolimits\!% \left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-5\pi i/6}\mathop{{% H^{(1)}_{-2/3}}\/}\nolimits\!\left(\zeta\right)+e^{5\pi i/6}\mathop{{H^{(2)}_{% -2/3}}\/}\nolimits\!\left(\zeta\right)\right),$
 9.6.8 $\mathop{\mathrm{Bi}\/}\nolimits\!\left(-z\right)=\sqrt{z/3}\left(\mathop{J_{-1% /3}\/}\nolimits\!\left(\zeta\right)-\mathop{J_{1/3}\/}\nolimits\!\left(\zeta% \right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{2\pi i/3}\mathop{{H^{(1)}_{1/3}}% \/}\nolimits\!\left(\zeta\right)+e^{-2\pi i/3}\mathop{{H^{(2)}_{1/3}}\/}% \nolimits\!\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/3}% \mathop{{H^{(1)}_{-1/3}}\/}\nolimits\!\left(\zeta\right)+e^{-\pi i/3}\mathop{{% H^{(2)}_{-1/3}}\/}\nolimits\!\left(\zeta\right)\right),$
 9.6.9 $\mathop{\mathrm{Bi}\/}\nolimits'\!\left(-z\right)=(z/\sqrt{3})\left(\mathop{J_% {-2/3}\/}\nolimits\!\left(\zeta\right)+\mathop{J_{2/3}\/}\nolimits\!\left(% \zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{\pi i/3}\mathop{{H^{(1)}_% {2/3}}\/}\nolimits\!\left(\zeta\right)+e^{-\pi i/3}\mathop{{H^{(2)}_{2/3}}\/}% \nolimits\!\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/3% }\mathop{{H^{(1)}_{-2/3}}\/}\nolimits\!\left(\zeta\right)+e^{\pi i/3}\mathop{{% H^{(2)}_{-2/3}}\/}\nolimits\!\left(\zeta\right)\right).$

§9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions

Again, for the notation see §§10.2(ii) and 10.25(ii). With

 9.6.10 $z=(\tfrac{3}{2}\zeta)^{2/3},$ Defines: $\zeta(z)$: change of variable (locally) Symbols: $z$: complex variable Permalink: http://dlmf.nist.gov/9.6.E10 Encodings: TeX, pMML, png See also: Annotations for 9.6(ii)
 9.6.11 $\displaystyle\mathop{J_{\pm 1/3}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{3/z}\left(\sqrt{3}\mathop{\mathrm{Ai}\/}% \nolimits\!\left(-z\right)\mp\mathop{\mathrm{Bi}\/}\nolimits\!\left(-z\right)% \right),$ 9.6.12 $\displaystyle\mathop{J_{\pm 2/3}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=\tfrac{1}{2}(\sqrt{3}/z)\left(\pm\sqrt{3}\mathop{\mathrm{Ai}\/}% \nolimits'\!\left(-z\right)+\mathop{\mathrm{Bi}\/}\nolimits'\!\left(-z\right)% \right),$
 9.6.13 $\displaystyle\mathop{I_{\pm 1/3}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{3/z}\left(\mp\sqrt{3}\mathop{\mathrm{Ai}\/}% \nolimits\!\left(z\right)+\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)% \right),$ 9.6.14 $\displaystyle\mathop{I_{\pm 2/3}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=\tfrac{1}{2}(\sqrt{3}/z)\left(\pm\sqrt{3}\mathop{\mathrm{Ai}\/}% \nolimits'\!\left(z\right)+\mathop{\mathrm{Bi}\/}\nolimits'\!\left(z\right)% \right),$
 9.6.15 $\displaystyle\mathop{K_{\pm 1/3}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=\pi\sqrt{3/z}\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right),$ 9.6.16 $\displaystyle\mathop{K_{\pm 2/3}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=-\pi(\sqrt{3}/z)\mathop{\mathrm{Ai}\/}\nolimits'\!\left(z\right),$
 9.6.17 $\displaystyle\mathop{{H^{(1)}_{1/3}}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=e^{-\pi i/3}\mathop{{H^{(1)}_{-1/3}}\/}\nolimits\!\left(\zeta% \right)=e^{-\pi i/6}\sqrt{3/z}\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(-z% \right)-i\mathop{\mathrm{Bi}\/}\nolimits\!\left(-z\right)\right),$ 9.6.18 $\displaystyle\mathop{{H^{(1)}_{2/3}}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=e^{-2\pi i/3}\mathop{{H^{(1)}_{-2/3}}\/}\nolimits\!\left(\zeta% \right)=e^{\pi i/6}(\sqrt{3}/z)\left(\mathop{\mathrm{Ai}\/}\nolimits'\!\left(-% z\right)-i\mathop{\mathrm{Bi}\/}\nolimits'\!\left(-z\right)\right),$
 9.6.19 $\displaystyle\mathop{{H^{(2)}_{1/3}}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=e^{\pi i/3}\mathop{{H^{(2)}_{-1/3}}\/}\nolimits\!\left(\zeta% \right)=e^{\pi i/6}\sqrt{3/z}\left(\mathop{\mathrm{Ai}\/}\nolimits\!\left(-z% \right)+i\mathop{\mathrm{Bi}\/}\nolimits\!\left(-z\right)\right),$ 9.6.20 $\displaystyle\mathop{{H^{(2)}_{2/3}}\/}\nolimits\!\left(\zeta\right)$ $\displaystyle=e^{2\pi i/3}\mathop{{H^{(2)}_{-2/3}}\/}\nolimits\!\left(\zeta% \right)=e^{-\pi i/6}(\sqrt{3}/z)\left(\mathop{\mathrm{Ai}\/}\nolimits'\!\left(% -z\right)+i\mathop{\mathrm{Bi}\/}\nolimits'\!\left(-z\right)\right).$

§9.6(iii) Airy Functions as Confluent Hypergeometric Functions

For the notation see §§13.1, 13.2, and 13.14(i). With $\zeta$ as in (9.6.1),

 9.6.21 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)$ $\displaystyle=\tfrac{1}{2}\pi^{-1/2}z^{-1/4}\mathop{W_{0,1/3}\/}\nolimits\!% \left(2\zeta\right)=3^{-1/6}\pi^{-1/2}\zeta^{2/3}e^{-\zeta}\mathop{U\/}% \nolimits\!\left(\tfrac{5}{6},\tfrac{5}{3},2\zeta\right),$ Defines: $\zeta(z)$: change of variable (locally) Symbols: $\mathop{\mathrm{Ai}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $\mathop{U\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$: Kummer confluent hypergeometric function, $\mathop{W_{\NVar{\kappa},\NVar{\mu}}\/}\nolimits\!\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of exponential function and $z$: complex variable Referenced by: §9.6(iii) Permalink: http://dlmf.nist.gov/9.6.E21 Encodings: TeX, pMML, png See also: Annotations for 9.6(iii) 9.6.22 $\displaystyle\mathop{\mathrm{Ai}\/}\nolimits'\!\left(z\right)$ $\displaystyle=-\tfrac{1}{2}\pi^{-1/2}z^{1/4}\mathop{W_{0,2/3}\/}\nolimits\!% \left(2\zeta\right)=-3^{1/6}\pi^{-1/2}\zeta^{4/3}e^{-\zeta}\mathop{U\/}% \nolimits\!\left(\tfrac{7}{6},\tfrac{7}{3},2\zeta\right),$
 9.6.23 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{1}{2^{1/3}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}% \right)}z^{-1/4}\mathop{M_{0,-1/3}\/}\nolimits\!\left(2\zeta\right)+\frac{3}{2% ^{5/3}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}\right)}z^{-1/4}\mathop{M_% {0,1/3}\/}\nolimits\!\left(2\zeta\right),$ 9.6.24 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits'\!\left(z\right)$ $\displaystyle=\frac{2^{1/3}}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}% \right)}z^{1/4}\mathop{M_{0,-2/3}\/}\nolimits\!\left(2\zeta\right)+\frac{3}{2^% {10/3}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}\right)}z^{1/4}\mathop{M_{% 0,2/3}\/}\nolimits\!\left(2\zeta\right),$
 9.6.25 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{1}{3^{1/6}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}% \right)}e^{-\zeta}\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\tfrac{1}{6};\tfrac% {1}{3};2\zeta\right)+\frac{3^{5/6}}{2^{2/3}\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{3}\right)}\zeta^{2/3}e^{-\zeta}\mathop{{{}_{1}F_{1}}\/}\nolimits\!% \left(\tfrac{5}{6};\tfrac{5}{3};2\zeta\right),$ 9.6.26 $\displaystyle\mathop{\mathrm{Bi}\/}\nolimits'\!\left(z\right)$ $\displaystyle=\frac{3^{1/6}}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}% \right)}e^{-\zeta}\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(-\tfrac{1}{6};-% \tfrac{1}{3};2\zeta\right)+\frac{3^{7/6}}{2^{7/3}\mathop{\Gamma\/}\nolimits\!% \left(\tfrac{2}{3}\right)}\zeta^{4/3}e^{-\zeta}\mathop{{{}_{1}F_{1}}\/}% \nolimits\!\left(\tfrac{7}{6};\tfrac{7}{3};2\zeta\right).$ Symbols: $\mathop{\mathrm{Bi}\/}\nolimits\!\left(\NVar{z}\right)$: Airy function, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\NVar{a};\NVar{b};\NVar{z}\right)$: $=\mathop{M\/}\nolimits\!\left(\NVar{a},\NVar{b},\NVar{z}\right)$ notation for the Kummer confluent hypergeometric function, $\mathrm{e}$: base of exponential function, $z$: complex variable and $\zeta(z)$: change of variable Referenced by: §9.6(iii), Equation (9.6.26) Permalink: http://dlmf.nist.gov/9.6.E26 Encodings: TeX, pMML, png Errata (effective with 1.0.9): Originally the second occurrence of the function $\mathop{{{}_{1}F_{1}}\/}\nolimits$ was given incorrectly as $\mathop{{{}_{1}F_{1}}\/}\nolimits\!\left(\tfrac{7}{6};\tfrac{7}{3};\zeta\right)$. Reported 2014-05-21 by Hanyou Chu See also: Annotations for 9.6(iii)

To express Airy functions in terms of hypergeometric functions combine §9.6(i) with (10.39.9).