scipy.special.airy¶
- scipy.special.airy(z) = <ufunc 'airy'>¶
Airy functions and their derivatives.
Parameters: z : array_like
Real or complex argument.
Returns: Ai, Aip, Bi, Bip : ndarrays
Airy functions Ai and Bi, and their derivatives Aip and Bip.
See also
- airye
- exponentially scaled Airy functions.
Notes
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 [R321] airy routine, which uses power series summation for small z and rational minimax approximations for large z.
Outside this range, the AMOS [R322] 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 \(t \equiv 2 z^{3/2}/3\)):
\[Ai(z) = \frac{1}{\pi \sqrt{3}} K_{1/3}(t)\]\[Ai'(z) = -\frac{z}{\pi \sqrt{3}} K_{2/3}(t)\]\[Bi(z) = \sqrt{\frac{z}{3}} \left(I_{-1/3}(t) + I_{1/3}(t) \right)\]\[Bi'(z) = \frac{z}{\sqrt{3}} \left(I_{-2/3}(t) + I_{2/3}(t)\right)\]References
[R321] (1, 2) Cephes Mathematical Functions Library, http://www.netlib.org/cephes/index.html [R322] (1, 2) Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/.org/amos/