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Module « scipy.special »

Fonction spherical_yn - module scipy.special

Signature de la fonction spherical_yn

def spherical_yn(n, z, derivative=False) 

Description

help(scipy.special.spherical_yn)

Spherical Bessel function of the second kind or its derivative.

Defined as [1]_,

.. math:: y_n(z) = \sqrt{\frac{\pi}{2z}} Y_{n + 1/2}(z),

where :math:`Y_n` is the Bessel function of the second kind.

Parameters
----------
n : int, array_like
    Order of the Bessel function (n >= 0).
z : complex or float, array_like
    Argument of the Bessel function.
derivative : bool, optional
    If True, the value of the derivative (rather than the function
    itself) is returned.

Returns
-------
yn : ndarray

Notes
-----
For real arguments, the function is computed using the ascending
recurrence [2]_.  For complex arguments, the definitional relation to
the cylindrical Bessel function of the second kind is used.

The derivative is computed using the relations [3]_,

.. math::
    y_n' = y_{n-1} - \frac{n + 1}{z} y_n.

    y_0' = -y_1


.. versionadded:: 0.18.0

References
----------
.. [1] https://dlmf.nist.gov/10.47.E4
.. [2] https://dlmf.nist.gov/10.51.E1
.. [3] https://dlmf.nist.gov/10.51.E2
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.

Examples
--------
The spherical Bessel functions of the second kind :math:`y_n` accept
both real and complex second argument. They can return a complex type:

>>> from scipy.special import spherical_yn
>>> spherical_yn(0, 3+5j)
(8.022343088587197-9.880052589376795j)
>>> type(spherical_yn(0, 3+5j))
<class 'numpy.complex128'>

We can verify the relation for the derivative from the Notes
for :math:`n=3` in the interval :math:`[1, 2]`:

>>> import numpy as np
>>> x = np.arange(1.0, 2.0, 0.01)
>>> np.allclose(spherical_yn(3, x, True),
...             spherical_yn(2, x) - 4/x * spherical_yn(3, x))
True

The first few :math:`y_n` with real argument:

>>> import matplotlib.pyplot as plt
>>> x = np.arange(0.0, 10.0, 0.01)
>>> fig, ax = plt.subplots()
>>> ax.set_ylim(-2.0, 1.0)
>>> ax.set_title(r'Spherical Bessel functions $y_n$')
>>> for n in np.arange(0, 4):
...     ax.plot(x, spherical_yn(n, x), label=rf'$y_{n}$')
>>> plt.legend(loc='best')
>>> plt.show()

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