Airy Functions

Watson's (1966, pp. 188-190) definition of an Airy function is the solution to the Airy Differential Equation

$$\Phi''\pm k^2 \Phi x =0$$ (1)

which is Finite at the Origin, where $\Phi'$ denotes the Derivative $d\Phi/dx$, $k^2=1/3$, and either Sign is permitted. Call these solutions $(1/\pi)\Phi(\pm k^2, x)$, then

$${1\over \pi}\Phi(\pm {\textstyle{1\over 3}}; x) \equiv \int_0^\infty \cos(t^3\pm xt)\,dt$$ (2)

$$\Phi({\textstyle{1\over 3}};x) = {\textstyle{1\over 3}}\pi \sqrt{{\textstyle{x\over 3}}} \left[{J_... ...^{3/2}\over 3^{3/2}}\right)+J_{1/3}\left({2x^{3/2}\over 3^{3/2}}\right)}\right]$$ (3)

$$\Phi(-{\textstyle{1\over 3}};x) = {\textstyle{1\over 3}}\pi \sqrt{{\textstyle{x\over 3}}} \left[{I_... ...{3/2}\over 3^{3/2}}\right)-I_{1/3}\left({2x^{3/2}\over 3^{3/2}}\right)}\right],$$ (4)

where $J(z)$ is a Bessel Function of the First Kind and $I(z)$ is a Modified Bessel Function of the First Kind. Using the identity

$$K_n(x)={\pi\over 2}{I_{-n}(x)-I_n(x)\over \sin(n\pi)},$$ (5)

where $K(z)$ is a Modified Bessel Function of the Second Kind, the second case can be re-expressed

$$\Phi(-{\textstyle{1\over 3}};x) = {\textstyle{1\over 3}}\pi \sqrt{{\textstyle{x\over 3}}} \left({2\... ...t({{\textstyle{1\over 3}}\pi}\right)K_{1/3}\left({2x^{3/2}\over 3^{3/2}}\right)$$ (6)

$$= {\pi\over 3}\sqrt{{\textstyle{1\over 3}}x}\, {2\over \pi} {\sqrt{3}\over 2} K_{1/3}\left({2x^{3/2}\over 3^{3/2}}\right)$$ (7)

$$= {\textstyle{1\over 3}}\sqrt{x}K_{1/3}\left({2x^{3/2}\over 3^{3/2}}\right).$$ (8)

A more commonly used definition of Airy functions is given by Abramowitz and Stegun (1972, pp. 446-447) and illustrated above. This definition identifies the $\mathop{\rm Ai}\nolimits (x)$ and $\mathop{\rm Bi}\nolimits (x)$ functions as the two Linearly Independent solutions to (1) with $k^2=1$ and a Minus Sign,

$$y''-yz=0.$$ (9)

The solutions are then written

$$y(z)=A\mathop{\rm Ai}\nolimits (z)+B\mathop{\rm Bi}\nolimits (z),$$ (10)

where

$$\mathop{\rm Ai}\nolimits (z) \equiv {1\over \pi}\Phi(-1,z)$$

$$= {\textstyle{1\over 3}}\sqrt{x}\,[I_{-1/3}({\textstyle{2\over 3}}z^{3/2})-I_{1/3}({\textstyle{2\over 3}}z^{3/2})]$$

$$= \sqrt{z \over {3\pi}}\, K_{1/3}({\textstyle{2\over 3}} z^{3/2})$$ (11)

$$\mathop{\rm Bi}\nolimits (z) \equiv \sqrt{z\over 3} \,[I_{-1/3}({\textstyle{2\over 3}}z^{3/2})+I_{1/3}({\textstyle{2\over 3}}z^{3/2})].$$ (12)

In the above plot, $\mathop{\rm Ai}\nolimits (z)$ is the solid curve and $\mathop{\rm Bi}\nolimits (z)$ is dashed. For zero argument,

$$\mathop{\rm Ai}\nolimits (0) = - {3^{-2/3}\over \Gamma({\textstyle{2\over 3}})},$$ (13)

where $\Gamma(z)$ is the Gamma Function. This means that Watson's expression becomes

$$(3a)^{-1/3}\pi\mathop{\rm Ai}\nolimits (\pm (3a)^{-1/3}x)=\int_0^\infty \cos(at^3\pm xt)\,dt.$$ (14)

A generalization has been constructed by Hardy.

The Asymptotic Series of $\mathop{\rm Ai}\nolimits (z)$ has a different form in different Quadrants of the Complex Plane, a fact known as the Stokes Phenomenon. Functions related to the Airy functions have been defined as

$$\mathop{\rm Gi}(z) \equiv {1\over \pi} \int_0^\infty \sin({\textstyle{1\over 3}} t^3+zt)\,dt$$ (15)

$$\mathop{\rm Hi}(z) \equiv {1\over \pi} \int_0^\infty {\rm exp}(-{\textstyle{1\over 3}} t^3+zt)\,dt.$$ (16)

See also Airy-Fock Functions

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Airy Functions.'' §10.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 446-452, 1972.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions.'' §6.7 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 234-245, 1992.

Spanier, J. and Oldham, K. B. ``The Airy Functions Ai($x$) and Bi($x$).'' Ch. 56 in An Atlas of Functions. Washington, DC: Hemisphere, pp. 555-562, 1987.

Watson, G. N. A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1966.