scipy.special.airy#

scipy.special.airy(z, out=None) = <ufunc 'airy'>#

Airy functions and their derivatives.

Parameters:

zarray_like: Real or complex argument.
outtuple of ndarray, optional: Optional output arrays for the function values

Returns:

Ai, Aip, Bi, Bip4-tuple of scalar or ndarray: Airy functions Ai and Bi, and their derivatives Aip and Bip.

See also

airye: exponentially scaled Airy functions.

Notes

The Airy functions \(\operatorname{Ai}\) and \(\operatorname{Bi}\) are two independent solutions of

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

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 \(t \equiv 2 z^{3/2}/3\)):

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

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/

Examples

Compute the Airy functions on the interval \([-15, 5]\).

>>> import numpy as np
>>> from scipy import special
>>> x = np.linspace(-15, 5, 201)
>>> ai, aip, bi, bip = special.airy(x)

Plot \(\operatorname{Ai}(x)\) and \(\operatorname{Bi}(x)\).

>>> import matplotlib.pyplot as plt
>>> plt.plot(x, ai, 'r', label='Ai(x)')
>>> plt.plot(x, bi, 'b--', label='Bi(x)')
>>> plt.ylim(-0.5, 1.0)
>>> plt.grid()
>>> plt.legend(loc='upper left')
>>> plt.show()