Module « scipy.special »

Fonction airye - module scipy.special

Signature de la fonction airye

Description

airye.__doc__

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

airye(z)

Exponentially scaled Airy functions and their derivatives.

Scaling::

    eAi  = Ai  * exp(2.0/3.0*z*sqrt(z))
    eAip = Aip * exp(2.0/3.0*z*sqrt(z))
    eBi  = Bi  * exp(-abs(2.0/3.0*(z*sqrt(z)).real))
    eBip = Bip * exp(-abs(2.0/3.0*(z*sqrt(z)).real))

Parameters
----------
z : array_like
    Real or complex argument.

Returns
-------
eAi, eAip, eBi, eBip : array_like
    Exponentially scaled Airy functions eAi and eBi, and their derivatives
    eAip and eBip

Notes
-----
Wrapper for the AMOS [1]_ routines `zairy` and `zbiry`.

See also
--------
airy

References
----------
.. [1] Donald E. Amos, "AMOS, A Portable Package for Bessel Functions
       of a Complex Argument and Nonnegative Order",
       http://netlib.org/amos/

Examples
--------
We can compute exponentially scaled Airy functions and their derivatives:

>>> from scipy.special import airye
>>> import matplotlib.pyplot as plt
>>> z = np.linspace(0, 50, 500)
>>> eAi, eAip, eBi, eBip = airye(z)
>>> f, ax = plt.subplots(2, 1, sharex=True)
>>> for ind, data in enumerate([[eAi, eAip, ["eAi", "eAip"]],
...                             [eBi, eBip, ["eBi", "eBip"]]]):
...     ax[ind].plot(z, data[0], "-r", z, data[1], "-b")
...     ax[ind].legend(data[2])
...     ax[ind].grid(True)
>>> plt.show()

We can compute these using usual non-scaled Airy functions by:

>>> from scipy.special import airy
>>> Ai, Aip, Bi, Bip = airy(z)
>>> np.allclose(eAi, Ai * np.exp(2.0 / 3.0 * z * np.sqrt(z)))
True
>>> np.allclose(eAip, Aip * np.exp(2.0 / 3.0 * z * np.sqrt(z)))
True
>>> np.allclose(eBi, Bi * np.exp(-abs(np.real(2.0 / 3.0 * z * np.sqrt(z)))))
True
>>> np.allclose(eBip, Bip * np.exp(-abs(np.real(2.0 / 3.0 * z * np.sqrt(z)))))
True

Comparing non-scaled and exponentially scaled ones, the usual non-scaled
function quickly underflows for large values, whereas the exponentially
scaled function does not.

>>> airy(200)
(0.0, 0.0, nan, nan)
>>> airye(200)
(0.07501041684381093, -1.0609012305109042, 0.15003188417418148, 2.1215836725571093)