scipy.special.airy#

scipy.special.airy(z, out=None) = <wrapped_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 \(t \equiv 2 z^{3/2}/3\)):

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

Array API Standard Support

airy has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	⛔
Dask	⛔	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. airy does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

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()