# mars.tensor.special.airy#

mars.tensor.special.airy(z, out=None, **kwargs)[source]#

Airy functions and their derivatives.

Parameters

z (array_like) – Real or complex argument.

Returns

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

Return type

ndarrays

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}

airye

exponentially scaled Airy functions.

References

1

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

2

Donald E. Amos, “AMOS, A Portable Package for Bessel Functions of a Complex Argument and Nonnegative Order”, http://netlib.org/amos/