9 Airy and Related FunctionsAiry Functions9.5 Integral Representations9.7 Asymptotic Expansions

§9.6 Relations to Other Functions

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Permalink:

http://dlmf.nist.gov/9.6

See also:

Annotations for Ch.9

Contents

1. §9.6(i) Airy Functions as Bessel Functions, Hankel Functions, and Modified Bessel Functions
2. §9.6(ii) Bessel Functions, Hankel Functions, and Modified Bessel Functions as Airy Functions
3. §9.6(iii) Airy Functions as Confluent Hypergeometric Functions

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

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Keywords:

Airy functions, Bessel functions, Hankel functions, modified Bessel functions, relations to other functions

Notes:

See Miller (1946, p. B17) and Olver (1997b, pp. 392–393).

Referenced by:

§10.16, §10.39, §10.72(i), §9.17(iv), §9.6(iii)

Permalink:

http://dlmf.nist.gov/9.6.i

See also:

Annotations for §9.6 and Ch.9

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

9.6.1		$$\zeta =\frac{2}{3}{z}^{3/2},$$
ⓘ 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		$\mathrm{Ai}\left(z\right)$	$={\pi }^{-1}\sqrt{z/3}{K}_{\pm 1/3}\left(\zeta \right)$
			$=\frac{1}{3}\sqrt{z}\left(I_{-1/3}\left(\zeta \right)-I_{1/3}\left(\zeta \right)\right)$
			$=\frac{1}{2}\sqrt{z/3}{\mathrm{e}}^{2\pi \mathrm{i}/3}{H}_{1/3}^{(1)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)$
			$=\frac{1}{2}\sqrt{z/3}{\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{-1/3}^{(1)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)$
			$=\frac{1}{2}\sqrt{z/3}{\mathrm{e}}^{-2\pi \mathrm{i}/3}{H}_{1/3}^{(2)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)$
			$=\frac{1}{2}\sqrt{z/3}{\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{-1/3}^{(2)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second 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: 10.4.14 (partial) Referenced by: (9.6.21) Permalink: http://dlmf.nist.gov/9.6.E2 Encodings: TeX, pMML, png See also: Annotations for §9.6(i), §9.6 and Ch.9
9.6.3		${\mathrm{Ai}}^{\prime }\left(z\right)$	$=-{\pi }^{-1}(z/\sqrt{3})K_{\pm 2/3}\left(\zeta \right)$
			$=(z/3)\left(I_{2/3}\left(\zeta \right)-I_{-2/3}\left(\zeta \right)\right)$
			$=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{2/3}^{(1)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)$
			$=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{-5\pi \mathrm{i}/6}{H}_{-2/3}^{(1)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)$
			$=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{2/3}^{(2)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)$
			$=\frac{1}{2}(z/\sqrt{3}){\mathrm{e}}^{5\pi \mathrm{i}/6}{H}_{-2/3}^{(2)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second 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: 10.4.16 (with opposite sign!) Referenced by: (9.6.22) Permalink: http://dlmf.nist.gov/9.6.E3 Encodings: TeX, pMML, png See also: Annotations for §9.6(i), §9.6 and Ch.9
9.6.4		$\mathrm{Bi}\left(z\right)$	$\begin{array}{ll}& =\sqrt{z/3}\left(I_{1/3}\left(\zeta \right)+I_{-1/3}\left(\zeta \right)\right)\\ & =\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{1/3}^{(1)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{1/3}^{(2)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)\right)\\ & =\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{-1/3}^{(1)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{-1/3}^{(2)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)\right),\end{array}$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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, $I_{\nu }\left(z\right)$: 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		${\mathrm{Bi}}^{\prime }\left(z\right)$	$\begin{array}{ll}& =(z/\sqrt{3})\left(I_{2/3}\left(\zeta \right)+I_{-2/3}\left(\zeta \right)\right)\\ & =\frac{1}{2}(z/\sqrt{3})\left({\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{2/3}^{(1)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{2/3}^{(2)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)\right)\\ & =\frac{1}{2}(z/\sqrt{3})\left({\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{-2/3}^{(1)}\left(\zeta {\mathrm{e}}^{-\pi \mathrm{i}/2}\right)+{\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{-2/3}^{(2)}\left(\zeta {\mathrm{e}}^{\pi \mathrm{i}/2}\right)\right),\end{array}$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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, $I_{\nu }\left(z\right)$: 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		$$\begin{array}{ll}\mathrm{Ai}\left(-z\right)& =(\sqrt{z}/3)\left(J_{1/3}\left(\zeta \right)+J_{-1/3}\left(\zeta \right)\right)=\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{1/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{1/3}^{(2)}\left(\zeta \right)\right)\\ & =\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{-1/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{-1/3}^{(2)}\left(\zeta \right)\right),\end{array}$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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: 10.4.15 (partial) Permalink: http://dlmf.nist.gov/9.6.E6 Encodings: TeX, pMML, png See also: Annotations for §9.6(i), §9.6 and Ch.9

9.6.7		$$\begin{array}{ll}{\mathrm{Ai}}^{\prime }\left(-z\right)& =(z/3)\left(J_{2/3}\left(\zeta \right)-J_{-2/3}\left(\zeta \right)\right)=\frac{1}{2}(z/\sqrt{3})\left({\mathrm{e}}^{-\pi \mathrm{i}/6}{H}_{2/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{\pi \mathrm{i}/6}{H}_{2/3}^{(2)}\left(\zeta \right)\right)\\ & =\frac{1}{2}(z/\sqrt{3})\left({\mathrm{e}}^{-5\pi \mathrm{i}/6}{H}_{-2/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{5\pi \mathrm{i}/6}{H}_{-2/3}^{(2)}\left(\zeta \right)\right),\end{array}$$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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: 10.4.17 (partial) Permalink: http://dlmf.nist.gov/9.6.E7 Encodings: TeX, pMML, png See also: Annotations for §9.6(i), §9.6 and Ch.9

9.6.8		$$\begin{array}{ll}\mathrm{Bi}\left(-z\right)& =\sqrt{z/3}\left(J_{-1/3}\left(\zeta \right)-J_{1/3}\left(\zeta \right)\right)\\ & =\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{2\pi \mathrm{i}/3}{H}_{1/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{-2\pi \mathrm{i}/3}{H}_{1/3}^{(2)}\left(\zeta \right)\right)\\ & =\frac{1}{2}\sqrt{z/3}\left({\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{-1/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{-1/3}^{(2)}\left(\zeta \right)\right),\end{array}$$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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		$$\begin{array}{ll}{\mathrm{Bi}}^{\prime }\left(-z\right)& =(z/\sqrt{3})\left(J_{-2/3}\left(\zeta \right)+J_{2/3}\left(\zeta \right)\right)=\frac{1}{2}(z/\sqrt{3})\left({\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{2/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{2/3}^{(2)}\left(\zeta \right)\right)\\ & =\frac{1}{2}(z/\sqrt{3})\left({\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{-2/3}^{(1)}\left(\zeta \right)+{\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{-2/3}^{(2)}\left(\zeta \right)\right).\end{array}$$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: Bessel function of the first kind, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: 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: 10.4.21 (partial) Permalink: http://dlmf.nist.gov/9.6.E9 Encodings: TeX, pMML, png See also: Annotations for §9.6(i), §9.6 and Ch.9

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

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Keywords:

Airy functions, Bessel functions, Hankel functions, modified Bessel functions, relations to other functions

Notes:

See Olver (1997b, pp. 392–393).

Referenced by:

§10.16, §10.39

Permalink:

http://dlmf.nist.gov/9.6.ii

See also:

Annotations for §9.6 and Ch.9

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

9.6.10		$$z={(\frac{3}{2}\zeta )}^{2/3},$$
ⓘ 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		$J_{\pm 1/3}\left(\zeta \right)$	$=\frac{1}{2}\sqrt{3/z}\left(\sqrt{3}\mathrm{Ai}\left(-z\right)\mp \mathrm{Bi}\left(-z\right)\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: 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		$J_{\pm 2/3}\left(\zeta \right)$	$=\frac{1}{2}(\sqrt{3}/z)\left(\pm \sqrt{3}{\mathrm{Ai}}^{\prime }\left(-z\right)+{\mathrm{Bi}}^{\prime }\left(-z\right)\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $J_{\nu }\left(z\right)$: 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		$I_{\pm 1/3}\left(\zeta \right)$	$=\frac{1}{2}\sqrt{3/z}\left(\mp \sqrt{3}\mathrm{Ai}\left(z\right)+\mathrm{Bi}\left(z\right)\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $I_{\nu }\left(z\right)$: 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		$I_{\pm 2/3}\left(\zeta \right)$	$=\frac{1}{2}(\sqrt{3}/z)\left(\pm \sqrt{3}{\mathrm{Ai}}^{\prime }\left(z\right)+{\mathrm{Bi}}^{\prime }\left(z\right)\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $I_{\nu }\left(z\right)$: 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		$K_{\pm 1/3}\left(\zeta \right)$	$=\pi \sqrt{3/z}\mathrm{Ai}\left(z\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: 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		$K_{\pm 2/3}\left(\zeta \right)$	$=-\pi (\sqrt{3}/z){\mathrm{Ai}}^{\prime }\left(z\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: 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		$H_{1/3}^{(1)}\left(\zeta \right)$	$={\mathrm{e}}^{-\pi \mathrm{i}/3}{H}_{-1/3}^{(1)}\left(\zeta \right)={\mathrm{e}}^{-\pi \mathrm{i}/6}\sqrt{3/z}\left(\mathrm{Ai}\left(-z\right)-\mathrm{i}\mathrm{Bi}\left(-z\right)\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: 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		$H_{2/3}^{(1)}\left(\zeta \right)$	$={\mathrm{e}}^{-2\pi \mathrm{i}/3}{H}_{-2/3}^{(1)}\left(\zeta \right)={\mathrm{e}}^{\pi \mathrm{i}/6}(\sqrt{3}/z)\left({\mathrm{Ai}}^{\prime }\left(-z\right)-\mathrm{i}{\mathrm{Bi}}^{\prime }\left(-z\right)\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: 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		$H_{1/3}^{(2)}\left(\zeta \right)$	$={\mathrm{e}}^{\pi \mathrm{i}/3}{H}_{-1/3}^{(2)}\left(\zeta \right)={\mathrm{e}}^{\pi \mathrm{i}/6}\sqrt{3/z}\left(\mathrm{Ai}\left(-z\right)+\mathrm{i}\mathrm{Bi}\left(-z\right)\right),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $H_{\nu }^{(2)}\left(z\right)$: 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		$H_{2/3}^{(2)}\left(\zeta \right)$	$={\mathrm{e}}^{2\pi \mathrm{i}/3}{H}_{-2/3}^{(2)}\left(\zeta \right)={\mathrm{e}}^{-\pi \mathrm{i}/6}(\sqrt{3}/z)\left({\mathrm{Ai}}^{\prime }\left(-z\right)+\mathrm{i}{\mathrm{Bi}}^{\prime }\left(-z\right)\right).$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $H_{\nu }^{(2)}\left(z\right)$: 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

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Keywords:

Airy functions, confluent hypergeometric functions, relations to other functions

Referenced by:

Erratum (V1.0.8) for Subsections 9.6(iii), 22.19(i)

Permalink:

http://dlmf.nist.gov/9.6.iii

Addition (effective with 1.0.8):

The final sentence in this subsection has been added.

Suggested 2014-01-27 by Tom Koornwinder

See also:

Annotations for §9.6 and Ch.9

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

9.6.21		$\mathrm{Ai}\left(z\right)$	$=\frac{1}{2}{\pi }^{-1/2}{z}^{-1/4}{W}_{0,1/3}\left(2\zeta \right)=3^{-1/6}{\pi }^{-1/2}{\zeta }^{2/3}{\mathrm{e}}^{-\zeta }U(\frac{5}{6},\frac{5}{3},2\zeta ),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable Proof sketch: Combine (9.6.2) with (10.39.8) and (10.39.6). Referenced by: (9.5.8) Permalink: http://dlmf.nist.gov/9.6.E21 Encodings: TeX, pMML, png See also: Annotations for §9.6(iii), §9.6 and Ch.9
9.6.22		${\mathrm{Ai}}^{\prime }\left(z\right)$	$=-\frac{1}{2}{\pi }^{-1/2}{z}^{1/4}{W}_{0,2/3}\left(2\zeta \right)=-3^{1/6}{\pi }^{-1/2}{\zeta }^{4/3}{\mathrm{e}}^{-\zeta }U(\frac{7}{6},\frac{7}{3},2\zeta ),$
ⓘ Symbols: $\mathrm{Ai}\left(z\right)$: Airy function, $U(a,b,z)$: Kummer confluent hypergeometric function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable Proof sketch: Combine (9.6.3) with (10.39.8) and (10.39.6). Permalink: http://dlmf.nist.gov/9.6.E22 Encodings: TeX, pMML, png See also: Annotations for §9.6(iii), §9.6 and Ch.9

9.6.23		$\mathrm{Bi}\left(z\right)$	$={\displaystyle \frac{1}{2^{1/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}}{z}^{-1/4}{M}_{0,-1/3}\left(2\zeta \right)+{\displaystyle \frac{3}{2^{5/3}\mathrm{\Gamma }\left(\frac{1}{3}\right)}}{z}^{-1/4}{M}_{0,1/3}\left(2\zeta \right),$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: 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		${\mathrm{Bi}}^{\prime }\left(z\right)$	$={\displaystyle \frac{2^{1/3}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}}{z}^{1/4}{M}_{0,-2/3}\left(2\zeta \right)+{\displaystyle \frac{3}{2^{10/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}}{z}^{1/4}{M}_{0,2/3}\left(2\zeta \right),$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: 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		$\mathrm{Bi}\left(z\right)$	$={\displaystyle \frac{1}{3^{1/6}\mathrm{\Gamma }\left(\frac{2}{3}\right)}}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(\frac{1}{6};\frac{1}{3};2\zeta )+{\displaystyle \frac{3^{5/6}}{2^{2/3}\mathrm{\Gamma }\left(\frac{1}{3}\right)}}{\zeta }^{2/3}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(\frac{5}{6};\frac{5}{3};2\zeta ),$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ notation for the Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\zeta$: change of variable Proof sketch: Combine (9.6.23) with (13.14.2) and refer to §13.1. Permalink: http://dlmf.nist.gov/9.6.E25 Encodings: TeX, pMML, png See also: Annotations for §9.6(iii), §9.6 and Ch.9
9.6.26		${\mathrm{Bi}}^{\prime }\left(z\right)$	$={\displaystyle \frac{3^{1/6}}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(-\frac{1}{6};-\frac{1}{3};2\zeta )+{\displaystyle \frac{3^{7/6}}{2^{7/3}\mathrm{\Gamma }\left(\frac{2}{3}\right)}}{\zeta }^{4/3}{\mathrm{e}}^{-\zeta }{}_1{}^{}F_1^{}(\frac{7}{6};\frac{7}{3};2\zeta ).$
ⓘ Symbols: $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_1{}^{}F_1^{}(a;b;z)$: $=M(a,b,z)$ 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 ${}_1{}^{}F_1^{}(\frac{7}{6};\frac{7}{3};\zeta )$. 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).

9.5 Integral Representations9.7 Asymptotic Expansions

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