maxframe.tensor.special.airy#

maxframe.tensor.special.airy(z, out=None)[source]#

Airy functions and their derivatives.

Parameters:

z (array_like) – Real or complex argument.
out (tuple of ndarray, optional) – Optional output arrays for the function values

Returns:

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

Return type:

4-tuple of scalar or ndarray

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 [1] airy routine, which uses power series summation for small z and rational minimax approximations for large z.

Outside this range, the AMOS [2] 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 \(t \equiv 2 z^{3/2}/3\)):

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

References