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Module « scipy.special »

Fonction spherical_jn - module scipy.special

Signature de la fonction spherical_jn

def spherical_jn(n, z, derivative=False) 

Description

help(scipy.special.spherical_jn)

Spherical Bessel function of the first kind or its derivative.

Defined as [1]_,

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

where :math:`J_n` is the Bessel function of the first 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
-------
jn : ndarray

Notes
-----
For real arguments greater than the order, the function is computed
using the ascending recurrence [2]_. For small real or complex
arguments, the definitional relation to the cylindrical Bessel function
of the first kind is used.

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

.. math::
    j_n'(z) = j_{n-1}(z) - \frac{n + 1}{z} j_n(z).

    j_0'(z) = -j_1(z)


.. versionadded:: 0.18.0

References
----------
.. [1] https://dlmf.nist.gov/10.47.E3
.. [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 first kind :math:`j_n` accept
both real and complex second argument. They can return a complex type:

>>> from scipy.special import spherical_jn
>>> spherical_jn(0, 3+5j)
(-9.878987731663194-8.021894345786002j)
>>> type(spherical_jn(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_jn(3, x, True),
...             spherical_jn(2, x) - 4/x * spherical_jn(3, x))
True

The first few :math:`j_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(-0.5, 1.5)
>>> ax.set_title(r'Spherical Bessel functions $j_n$')
>>> for n in np.arange(0, 4):
...     ax.plot(x, spherical_jn(n, x), label=rf'$j_{n}$')
>>> plt.legend(loc='best')
>>> plt.show()

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