# §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: $\zeta$: change of variable (locally) Symbols: $z$: complex variable Referenced by: §9.6(iii), (9.7.17), §9.7(iv) Permalink: http://dlmf.nist.gov/9.6.E1 Encodings: TeX, pMML, png See also: Annotations for §9.6(i), §9.6 and Ch.9
 9.6.2 $\displaystyle\operatorname{Ai}\left(z\right)$ $\displaystyle=\pi^{-1}\sqrt{z/3}K_{\pm 1/3}\left(\zeta\right)$ $\displaystyle=\tfrac{1}{3}\sqrt{z}\left(I_{-1/3}\left(\zeta\right)-I_{1/3}% \left(\zeta\right)\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{z/3}e^{2\pi i/3}{H^{(1)}_{1/3}}\left(\zeta e^{% \pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{z/3}e^{\pi i/3}{H^{(1)}_{-1/3}}\left(\zeta e^{% \pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{z/3}e^{-2\pi i/3}{H^{(2)}_{1/3}}\left(\zeta e^% {-\pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{z/3}e^{-\pi i/3}{H^{(2)}_{-1/3}}\left(\zeta e^% {-\pi i/2}\right),$ 9.6.3 $\displaystyle\operatorname{Ai}'\left(z\right)$ $\displaystyle=-\pi^{-1}(z/\sqrt{3})K_{\pm 2/3}\left(\zeta\right)$ $\displaystyle=(z/3)\left(I_{2/3}\left(\zeta\right)-I_{-2/3}\left(\zeta\right)\right)$ $\displaystyle=\tfrac{1}{2}(z/\sqrt{3})e^{-\pi i/6}{H^{(1)}_{2/3}}\left(\zeta e% ^{\pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}(z/\sqrt{3})e^{-5\pi i/6}{H^{(1)}_{-2/3}}\left(\zeta e% ^{\pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}(z/\sqrt{3})e^{\pi i/6}{H^{(2)}_{2/3}}\left(\zeta e^% {-\pi i/2}\right)$ $\displaystyle=\tfrac{1}{2}(z/\sqrt{3})e^{5\pi i/6}{H^{(2)}_{-2/3}}\left(\zeta e% ^{-\pi i/2}\right),$ 9.6.4 $\displaystyle\operatorname{Bi}\left(z\right)$ $\displaystyle=\sqrt{z/3}\left(I_{1/3}\left(\zeta\right)+I_{-1/3}\left(\zeta% \right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/6}{H^{(1)}_{1/3}}\left(% \zeta e^{-\pi i/2}\right)+e^{-\pi i/6}{H^{(2)}_{1/3}}\left(\zeta e^{\pi i/2}% \right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{-\pi i/6}{H^{(1)}_{-1/3}}\left(% \zeta e^{-\pi i/2}\right)+e^{\pi i/6}{H^{(2)}_{-1/3}}\left(\zeta e^{\pi i/2}% \right)\right),$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $z$: complex variable and $\zeta$: change of variable Proof sketch: Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393). A&S Ref: 10.4.18 (partial) Referenced by: (9.6.23), (9.7.16), §9.7(iii) Permalink: http://dlmf.nist.gov/9.6.E4 Encodings: TeX, pMML, png See also: Annotations for §9.6(i), §9.6 and Ch.9 9.6.5 $\displaystyle\operatorname{Bi}'\left(z\right)$ $\displaystyle=(z/\sqrt{3})\left(I_{2/3}\left(\zeta\right)+I_{-2/3}\left(\zeta% \right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{\pi i/3}{H^{(1)}_{2/3}}\left(% \zeta e^{-\pi i/2}\right)+e^{-\pi i/3}{H^{(2)}_{2/3}}\left(\zeta e^{\pi i/2}% \right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/3}{H^{(1)}_{-2/3}}\left% (\zeta e^{-\pi i/2}\right)+e^{\pi i/3}{H^{(2)}_{-2/3}}\left(\zeta e^{\pi i/2}% \right)\right),$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $z$: complex variable and $\zeta$: change of variable Proof sketch: Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393). A&S Ref: 10.4.20 (partial) Referenced by: (9.6.24), (9.7.16), (9.7.17), §9.7(iii), §9.7(iv) Permalink: http://dlmf.nist.gov/9.6.E5 Encodings: TeX, pMML, png See also: Annotations for §9.6(i), §9.6 and Ch.9
 9.6.6 $\operatorname{Ai}\left(-z\right)=(\sqrt{z}/3)\left(J_{1/3}\left(\zeta\right)+J% _{-1/3}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/6}{H^{(1% )}_{1/3}}\left(\zeta\right)+e^{-\pi i/6}{H^{(2)}_{1/3}}\left(\zeta\right)% \right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{-\pi i/6}{H^{(1)}_{-1/3}}\left(\zeta% \right)+e^{\pi i/6}{H^{(2)}_{-1/3}}\left(\zeta\right)\right),$
 9.6.7 $\operatorname{Ai}'\left(-z\right)=(z/3)\left(J_{2/3}\left(\zeta\right)-J_{-2/3% }\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/6}{H^{(1)}_% {2/3}}\left(\zeta\right)+e^{\pi i/6}{H^{(2)}_{2/3}}\left(\zeta\right)\right)=% \tfrac{1}{2}(z/\sqrt{3})\left(e^{-5\pi i/6}{H^{(1)}_{-2/3}}\left(\zeta\right)+% e^{5\pi i/6}{H^{(2)}_{-2/3}}\left(\zeta\right)\right),$
 9.6.8 $\operatorname{Bi}\left(-z\right)=\sqrt{z/3}\left(J_{-1/3}\left(\zeta\right)-J_% {1/3}\left(\zeta\right)\right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{2\pi i/3}{H^{(1)% }_{1/3}}\left(\zeta\right)+e^{-2\pi i/3}{H^{(2)}_{1/3}}\left(\zeta\right)% \right)=\tfrac{1}{2}\sqrt{z/3}\left(e^{\pi i/3}{H^{(1)}_{-1/3}}\left(\zeta% \right)+e^{-\pi i/3}{H^{(2)}_{-1/3}}\left(\zeta\right)\right),$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta$: change of variable Proof sketch: Derivable from those given in Miller (1946, p. B17) and Olver (1997b, pp. 392–393). A&S Ref: 10.4.19 (in slightly different form) Permalink: http://dlmf.nist.gov/9.6.E8 Encodings: TeX, pMML, png See also: Annotations for §9.6(i), §9.6 and Ch.9
 9.6.9 $\operatorname{Bi}'\left(-z\right)=(z/\sqrt{3})\left(J_{-2/3}\left(\zeta\right)% +J_{2/3}\left(\zeta\right)\right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{\pi i/3}{H^% {(1)}_{2/3}}\left(\zeta\right)+e^{-\pi i/3}{H^{(2)}_{2/3}}\left(\zeta\right)% \right)=\tfrac{1}{2}(z/\sqrt{3})\left(e^{-\pi i/3}{H^{(1)}_{-2/3}}\left(\zeta% \right)+e^{\pi i/3}{H^{(2)}_{-2/3}}\left(\zeta\right)\right).$

## §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 Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). Permalink: http://dlmf.nist.gov/9.6.E10 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9
 9.6.11 $\displaystyle J_{\pm 1/3}\left(\zeta\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{3/z}\left(\sqrt{3}\operatorname{Ai}\left(-z% \right)\mp\operatorname{Bi}\left(-z\right)\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $z$: complex variable and $\zeta(z)$: change of variable Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). A&S Ref: 10.4.22 Permalink: http://dlmf.nist.gov/9.6.E11 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9 9.6.12 $\displaystyle J_{\pm 2/3}\left(\zeta\right)$ $\displaystyle=\tfrac{1}{2}(\sqrt{3}/z)\left(\pm\sqrt{3}\operatorname{Ai}'\left% (-z\right)+\operatorname{Bi}'\left(-z\right)\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $J_{\NVar{\nu}}\left(\NVar{z}\right)$: Bessel function of the first kind, $z$: complex variable and $\zeta(z)$: change of variable Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). A&S Ref: 10.4.27 Permalink: http://dlmf.nist.gov/9.6.E12 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9
 9.6.13 $\displaystyle I_{\pm 1/3}\left(\zeta\right)$ $\displaystyle=\tfrac{1}{2}\sqrt{3/z}\left(\mp\sqrt{3}\operatorname{Ai}\left(z% \right)+\operatorname{Bi}\left(z\right)\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $z$: complex variable and $\zeta(z)$: change of variable Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). A&S Ref: 10.4.25 Permalink: http://dlmf.nist.gov/9.6.E13 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9 9.6.14 $\displaystyle I_{\pm 2/3}\left(\zeta\right)$ $\displaystyle=\tfrac{1}{2}(\sqrt{3}/z)\left(\pm\sqrt{3}\operatorname{Ai}'\left% (z\right)+\operatorname{Bi}'\left(z\right)\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the first kind, $z$: complex variable and $\zeta(z)$: change of variable Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). A&S Ref: 10.4.30 Permalink: http://dlmf.nist.gov/9.6.E14 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9
 9.6.15 $\displaystyle K_{\pm 1/3}\left(\zeta\right)$ $\displaystyle=\pi\sqrt{3/z}\operatorname{Ai}\left(z\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind, $z$: complex variable and $\zeta(z)$: change of variable Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). A&S Ref: 10.4.26 Permalink: http://dlmf.nist.gov/9.6.E15 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9 9.6.16 $\displaystyle K_{\pm 2/3}\left(\zeta\right)$ $\displaystyle=-\pi(\sqrt{3}/z)\operatorname{Ai}'\left(z\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind, $z$: complex variable and $\zeta(z)$: change of variable Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). A&S Ref: 10.4.31 Permalink: http://dlmf.nist.gov/9.6.E16 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9
 9.6.17 $\displaystyle{H^{(1)}_{1/3}}\left(\zeta\right)$ $\displaystyle=e^{-\pi i/3}{H^{(1)}_{-1/3}}\left(\zeta\right)=e^{-\pi i/6}\sqrt% {3/z}\left(\operatorname{Ai}\left(-z\right)-i\operatorname{Bi}\left(-z\right)% \right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta(z)$: change of variable Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). A&S Ref: 10.4.23 (with different sign, different form!) Referenced by: (9.8.11), (9.8.9) Permalink: http://dlmf.nist.gov/9.6.E17 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9 9.6.18 $\displaystyle{H^{(1)}_{2/3}}\left(\zeta\right)$ $\displaystyle=e^{-2\pi i/3}{H^{(1)}_{-2/3}}\left(\zeta\right)=e^{\pi i/6}(% \sqrt{3}/z)\left(\operatorname{Ai}'\left(-z\right)-i\operatorname{Bi}'\left(-z% \right)\right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta(z)$: change of variable Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). A&S Ref: 10.4.28 Referenced by: (9.8.10), (9.8.12) Permalink: http://dlmf.nist.gov/9.6.E18 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9
 9.6.19 $\displaystyle{H^{(2)}_{1/3}}\left(\zeta\right)$ $\displaystyle=e^{\pi i/3}{H^{(2)}_{-1/3}}\left(\zeta\right)=e^{\pi i/6}\sqrt{3% /z}\left(\operatorname{Ai}\left(-z\right)+i\operatorname{Bi}\left(-z\right)% \right),$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta(z)$: change of variable Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). A&S Ref: 10.4.24 Referenced by: (9.8.11), (9.8.9) Permalink: http://dlmf.nist.gov/9.6.E19 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9 9.6.20 $\displaystyle{H^{(2)}_{2/3}}\left(\zeta\right)$ $\displaystyle=e^{2\pi i/3}{H^{(2)}_{-2/3}}\left(\zeta\right)=e^{-\pi i/6}(% \sqrt{3}/z)\left(\operatorname{Ai}'\left(-z\right)+i\operatorname{Bi}'\left(-z% \right)\right).$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$: Bessel function of the third kind (or Hankel function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $z$: complex variable and $\zeta(z)$: change of variable Proof sketch: Derivable from those given in Olver (1997b, pp. 392–393). A&S Ref: 10.4.29 Referenced by: (9.8.10), (9.8.12) Permalink: http://dlmf.nist.gov/9.6.E20 Encodings: TeX, pMML, png See also: Annotations for §9.6(ii), §9.6 and Ch.9

## §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 $\displaystyle\operatorname{Ai}\left(z\right)$ $\displaystyle=\tfrac{1}{2}\pi^{-1/2}z^{-1/4}W_{0,1/3}\left(2\zeta\right)=3^{-1% /6}\pi^{-1/2}\zeta^{2/3}e^{-\zeta}U\left(\tfrac{5}{6},\tfrac{5}{3},2\zeta% \right),$ 9.6.22 $\displaystyle\operatorname{Ai}'\left(z\right)$ $\displaystyle=-\tfrac{1}{2}\pi^{-1/2}z^{1/4}W_{0,2/3}\left(2\zeta\right)=-3^{1% /6}\pi^{-1/2}\zeta^{4/3}e^{-\zeta}U\left(\tfrac{7}{6},\tfrac{7}{3},2\zeta% \right),$
 9.6.23 $\displaystyle\operatorname{Bi}\left(z\right)$ $\displaystyle=\frac{1}{2^{1/3}\Gamma\left(\tfrac{2}{3}\right)}z^{-1/4}M_{0,-1/% 3}\left(2\zeta\right)+\frac{3}{2^{5/3}\Gamma\left(\tfrac{1}{3}\right)}z^{-1/4}% M_{0,1/3}\left(2\zeta\right),$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $\Gamma\left(\NVar{z}\right)$: gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $\zeta$: change of variable Proof sketch: Combine (9.6.4) with (10.39.7). Referenced by: (9.6.25) Permalink: http://dlmf.nist.gov/9.6.E23 Encodings: TeX, pMML, png See also: Annotations for §9.6(iii), §9.6 and Ch.9 9.6.24 $\displaystyle\operatorname{Bi}'\left(z\right)$ $\displaystyle=\frac{2^{1/3}}{\Gamma\left(\tfrac{1}{3}\right)}z^{1/4}M_{0,-2/3}% \left(2\zeta\right)+\frac{3}{2^{10/3}\Gamma\left(\tfrac{2}{3}\right)}z^{1/4}M_% {0,2/3}\left(2\zeta\right),$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $\Gamma\left(\NVar{z}\right)$: gamma function, $M_{\NVar{\kappa},\NVar{\mu}}\left(\NVar{z}\right)$: Whittaker confluent hypergeometric function, $z$: complex variable and $\zeta$: change of variable Proof sketch: Combine (9.6.5) with (10.39.7). Referenced by: (9.6.26) Permalink: http://dlmf.nist.gov/9.6.E24 Encodings: TeX, pMML, png See also: Annotations for §9.6(iii), §9.6 and Ch.9
 9.6.25 $\displaystyle\operatorname{Bi}\left(z\right)$ $\displaystyle=\frac{1}{3^{1/6}\Gamma\left(\tfrac{2}{3}\right)}e^{-\zeta}{{}_{1% }F_{1}}\left(\tfrac{1}{6};\tfrac{1}{3};2\zeta\right)+\frac{3^{5/6}}{2^{2/3}% \Gamma\left(\tfrac{1}{3}\right)}\zeta^{2/3}e^{-\zeta}{{}_{1}F_{1}}\left(\tfrac% {5}{6};\tfrac{5}{3};2\zeta\right),$ 9.6.26 $\displaystyle\operatorname{Bi}'\left(z\right)$ $\displaystyle=\frac{3^{1/6}}{\Gamma\left(\tfrac{1}{3}\right)}e^{-\zeta}{{}_{1}% F_{1}}\left(-\tfrac{1}{6};-\tfrac{1}{3};2\zeta\right)+\frac{3^{7/6}}{2^{7/3}% \Gamma\left(\tfrac{2}{3}\right)}\zeta^{4/3}e^{-\zeta}{{}_{1}F_{1}}\left(\tfrac% {7}{6};\tfrac{7}{3};2\zeta\right).$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: 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, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable Proof sketch: Combine (9.6.24) with (13.14.2) and refer to §13.1. Referenced by: Erratum (V1.0.9) for 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 ${{}_{1}F_{1}}$ was given incorrectly as ${{}_{1}F_{1}}\left(\tfrac{7}{6};\tfrac{7}{3};\zeta\right)$. Reported 2014-05-21 by Hanyou Chu See also: Annotations for §9.6(iii), §9.6 and Ch.9

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