scipy.special.airy(z) = <ufunc 'airy'>

Airy functions and their derivatives.


z : array_like

Real or complex argument.


Ai, Aip, Bi, Bip : ndarrays

Airy functions Ai and Bi, and their derivatives Aip and Bip.

See also

exponentially scaled Airy functions.


The Airy functions Ai and Bi are two independent solutions of

\[y''(x) = x y(x).\]

For real z in [-10, 10], the computation is carried out by calling the Cephes [R334] airy routine, which uses power series summation for small z and rational minimax approximations for large z.

Outside this range, the AMOS [R335] zairy and zbiry routines are employed. They are computed using power series for \(|z| < 1\) and the following relations to modified Bessel functions for larger z (where \(t \equiv 2 z^{3/2}/3\)):

\[Ai(z) = \frac{1}{\pi \sqrt{3}} K_{1/3}(t)\]\[Ai'(z) = -\frac{z}{\pi \sqrt{3}} K_{2/3}(t)\]\[Bi(z) = \sqrt{\frac{z}{3}} \left(I_{-1/3}(t) + I_{1/3}(t) \right)\]\[Bi'(z) = \frac{z}{\sqrt{3}} \left(I_{-2/3}(t) + I_{2/3}(t)\right)\]


[R334](1, 2) Cephes Mathematical Functions Library,
[R335](1, 2) Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”,