relation to Airy function

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1: 9.6 Relations to Other Functions

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

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

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9.6.20 $$H_{2/3}^{(2)}\left(\zeta \right)={\mathrm{e}}^{2\pi \mathrm{i}/3}{H}_{-2/3}^{(2)}\left(\zeta \right)={\mathrm{e}}^{-\pi \mathrm{i}/6}(\sqrt{3}/z)\left({\mathrm{Ai}}^{\prime }\left(-z\right)+\mathrm{i}{\mathrm{Bi}}^{\prime }\left(-z\right)\right).$$

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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:

pMML, png, TeX

See also:

Annotations for §9.6(ii), §9.6 and Ch.9

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§9.6(iii) Airy Functions as Confluent Hypergeometric Functions

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2: 13.18 Relations to Other Functions

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§13.18(iii) Modified Bessel Functions

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3: 13.6 Relations to Other Functions

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§13.6(iii) Modified Bessel Functions

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4: 36.2 Catastrophes and Canonical Integrals

… ► ${\mathrm{\Psi }}_1$ is related to the Airy function (§9.2): … ►Addendum: For further special cases see §36.2(iv) …

5: 9.9 Zeros

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§9.9(ii) Relation to Modulus and Phase

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6: 9.13 Generalized Airy Functions

… ►Swanson and Headley (1967) define independent solutions $A_n\left(z\right)$ and $B_n\left(z\right)$ of (9.13.1) by … ►

$$-A_n^{\prime }\left(0\right)=p^{1/2}{B}_n^{\prime }\left(0\right)=\frac{p^p}{\mathrm{\Gamma }\left(p\right)}.$$

… ►Their relations to the functions $A_n\left(z\right)$ and $B_n\left(z\right)$ are given by …

7: 9.16 Physical Applications

… ►The use of Airy function and related uniform asymptotic techniques to calculate amplitudes of polarized rainbows can be found in Nussenzveig (1992) and Adam (2002). …

8: 9.8 Modulus and Phase

… ►(These definitions of $\theta \left(x\right)$ and $\varphi \left(x\right)$ differ from Abramowitz and Stegun (1964, Chapter 10), and agree more closely with those used in Miller (1946) and Olver (1997b, Chapter 11).) …

9: Philip J. Davis

… ►After receiving an overview of the project and watching a short demo that included a few preliminary colorful, but static, 3D graphs constructed for the first Chapter, “Airy and Related Functions”, written by Olver, Davis expressed the hope that designing a web-based resource would allow the team to incorporate interesting computer graphics, such as function surfaces that could be rotated and examined. …

10: 10.16 Relations to Other Functions

§10.16 Relations to Other Functions

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Elementary Functions

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Airy Functions

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Parabolic Cylinder Functions

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Confluent Hypergeometric Functions

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