Airy function

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11: Sidebar 9.SB2: Interference Patterns in Caustics

… ►The oscillating intensity of the interference fringes across the caustic is described by the Airy function.

12: 9.13 Generalized Airy Functions

§9.13 Generalized Airy Functions

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§9.13(i) Generalizations from the Differential Equation

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§9.13(ii) Generalizations from Integral Representations

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13: 9.10 Integrals

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§9.10(i) Indefinite Integrals

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§9.10(ii) Asymptotic Approximations

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§9.10(iv) Definite Integrals

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§9.10(v) Laplace Transforms

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§9.10(vi) Mellin Transform

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14: 9.17 Methods of Computation

§9.17 Methods of Computation

… ►Among the integral representations of the Airy functions the Stieltjes transform (9.10.18) furnishes a way of computing $\mathrm{Ai}\left(z\right)$ in the complex plane, once values of this function can be generated on the positive real axis. … ►

§9.17(v) Zeros

►Zeros of the Airy functions, and their derivatives, can be computed to high precision via Newton’s rule (§3.8(ii)) or Halley’s rule (§3.8(v)), using values supplied by the asymptotic expansions of §9.9(iv) as initial approximations. … ►For the computation of the zeros of the Scorer functions and their derivatives see Gil et al. (2003c).

15: 9.12 Scorer Functions

… ►

9.12.4 $$\mathrm{Gi}\left(z\right)=\mathrm{Bi}\left(z\right){\int }_z^{\mathrm{\infty }}\mathrm{Ai}\left(t\right)dt+\mathrm{Ai}\left(z\right){\int }_0^z\mathrm{Bi}\left(t\right)dt,$$

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Defines:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Source:

Olver (1997b, (12.08), p. 430)

Referenced by:

(9.10.1)

Permalink:

http://dlmf.nist.gov/9.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(i), §9.12 and Ch.9

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9.12.5 $$\mathrm{Hi}\left(z\right)=\mathrm{Bi}\left(z\right){\int }_{-\mathrm{\infty }}^z\mathrm{Ai}\left(t\right)dt-\mathrm{Ai}\left(z\right){\int }_{-\mathrm{\infty }}^z\mathrm{Bi}\left(t\right)dt.$$

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Defines:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Source:

Olver (1997b, (12.09), p. 430)

Referenced by:

(9.10.3)

Permalink:

http://dlmf.nist.gov/9.12.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(i), §9.12 and Ch.9

… ► $-\mathrm{Gi}\left(x\right)$ is a numerically satisfactory companion to the complementary functions $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ on the interval . … ►

9.12.8 $$-\mathrm{Gi}\left(z\right),\mathrm{Ai}\left(z\right),\mathrm{Bi}\left(z\right),$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi$,

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase and $z$: complex variable

Source:

Olver (1997b, p. 432)

Permalink:

http://dlmf.nist.gov/9.12.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(iv), §9.12 and Ch.9

… ►

9.12.11 $$\mathrm{Gi}\left(z\right)+\mathrm{Hi}\left(z\right)=\mathrm{Bi}\left(z\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function) and $z$: complex variable

Proof sketch:

Verify with the aid of §§9.2(ii) and 9.12(iii).

Referenced by:

(9.10.2), (9.12.15), (9.12.19), §9.12(viii)

Permalink:

http://dlmf.nist.gov/9.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(v), §9.12 and Ch.9

…

16: 9.4 Maclaurin Series

§9.4 Maclaurin Series

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9.4.1 $$\mathrm{Ai}\left(z\right)=\mathrm{Ai}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Ai}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Source:

Olver (1997b, p. 54)

A&S Ref:

10.4.2 (with 10.4.4 and 10.4.5)

Permalink:

http://dlmf.nist.gov/9.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

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9.4.2 $${\mathrm{Ai}}^{\prime }\left(z\right)={\mathrm{Ai}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Ai}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

►

9.4.3 $$\mathrm{Bi}\left(z\right)=\mathrm{Bi}\left(0\right)\left(1+\frac{1}{3!}{z}^3+\frac{1\cdot 4}{6!}{z}^6+\frac{1\cdot 4\cdot 7}{9!}{z}^9+\mathrm{\cdots }\right)+{\mathrm{Bi}}^{\prime }\left(0\right)\left(z+\frac{2}{4!}{z}^4+\frac{2\cdot 5}{7!}{z}^7+\frac{2\cdot 5\cdot 8}{10!}{z}^{10}+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

A&S Ref:

10.4.3 (with 10.4.4 and 10.4.5)

Referenced by:

(9.12.15)

Permalink:

http://dlmf.nist.gov/9.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

►

9.4.4 $${\mathrm{Bi}}^{\prime }\left(z\right)={\mathrm{Bi}}^{\prime }\left(0\right)\left(1+\frac{2}{3!}{z}^3+\frac{2\cdot 5}{6!}{z}^6+\frac{2\cdot 5\cdot 8}{9!}{z}^9+\mathrm{\cdots }\right)+\mathrm{Bi}\left(0\right)\left(\frac{1}{2!}{z}^2+\frac{1\cdot 4}{5!}{z}^5+\frac{1\cdot 4\cdot 7}{8!}{z}^8+\mathrm{\cdots }\right).$$

ⓘ

Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $!$: factorial (as in $n!$) and $z$: complex variable

Proof sketch:

Use methods of Olver (1997b, p. 54).

Permalink:

http://dlmf.nist.gov/9.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §9.4 and Ch.9

17: 9.9 Zeros

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§9.9(ii) Relation to Modulus and Phase

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§9.9(iii) Derivatives With Respect to $k$

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§9.9(iv) Asymptotic Expansions

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§9.9(v) Tables

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Table 9.9.4: Complex zeros of ${\mathrm{Bi}}^{\prime }$.

► ►►

	${\mathrm{e}}^{-\pi \mathrm{i}/3}{\beta }_k^{\prime }$		$\mathrm{Bi}\left({\beta }_k^{\prime }\right)$
…

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18: 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

►

§9.6(iii) Airy Functions as Confluent Hypergeometric Functions

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

19: 9.5 Integral Representations

… ►

§9.5(i) Real Variable

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9.5.1 $$\mathrm{Ai}\left(x\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\cos \left(\frac{1}{3}{t}^3+xt\right)dt.$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: 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:

pMML, png, TeX

See also:

Annotations for §9.5(i), §9.5 and Ch.9

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9.5.2 $$\mathrm{Ai}\left(-x\right)=\frac{x^{1/2}}{\pi }{\int }_{-1}^{\mathrm{\infty }}\cos \left(x^{3/2}(\frac{1}{3}{t}^3+t^2-\frac{2}{3})\right)dt,$$ $x>0$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: 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:

pMML, png, TeX

See also:

Annotations for §9.5(i), §9.5 and Ch.9

… ►

§9.5(ii) Complex Variable

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9.5.6 $$\mathrm{Ai}\left(z\right)=\frac{\sqrt{3}}{2\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{t^3}{3}-\frac{z^3}{3t^3}\right)dt,$$ .

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $\mathrm{ph}$: 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:

pMML, png, TeX

Clarification (effective with 1.0.15):

The validity constraint was added.

See also:

Annotations for §9.5(ii), §9.5 and Ch.9

…

20: 32.5 Integral Equations

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32.5.1 $$K(z,\zeta )=k\mathrm{Ai}\left(\frac{z+\zeta }{2}\right)+\frac{k^2}{4}{\int }_z^{\mathrm{\infty }}{\int }_z^{\mathrm{\infty }}K(z,s)\mathrm{Ai}\left(\frac{s+t}{2}\right)\mathrm{Ai}\left(\frac{t+\zeta }{2}\right)dsdt,$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $dx$: differential of $x$ , $\int$: integral, $z$: real, $k$: real and $K(z,\zeta )$: solution

Permalink:

http://dlmf.nist.gov/32.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32

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32.5.3 $$w(z)\sim k\mathrm{Ai}\left(z\right),$$ $z\to +\mathrm{\infty }$.

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: asymptotic equality, $z$: real and $k$: real

Permalink:

http://dlmf.nist.gov/32.5.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §32.5 and Ch.32