Module « scipy.special »

Fonction airy - module scipy.special

Signature de la fonction airy

Description

airy.__doc__

airy(x[, out1, out2, out3, out4], / [, out=(None, None, None, None)], *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature, extobj])

airy(z)

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.

Notes
-----
The Airy functions Ai and Bi are two independent solutions of

.. math:: 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 :math:`|z| < 1` and
the following relations to modified Bessel functions for larger `z`
(where :math:`t \equiv 2 z^{3/2}/3`):

.. math::

    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)

See also
--------
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/

Examples
--------
Compute the Airy functions on the interval [-15, 5].

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

Plot Ai(x) and 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()