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Module « scipy.special »

Fonction spherical_in - module scipy.special

Signature de la fonction spherical_in

def spherical_in(n, z, derivative=False) 

Description

help(scipy.special.spherical_in)

Modified spherical Bessel function of the first kind or its derivative.

Defined as [1]_,

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

where :math:`I_n` is the modified 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
-------
in : ndarray

Notes
-----
The function is computed using its definitional relation to the
modified cylindrical Bessel function of the first kind.

The derivative is computed using the relations [2]_,

.. math::
    i_n' = i_{n-1} - \frac{n + 1}{z} i_n.

    i_1' = i_0


.. versionadded:: 0.18.0

References
----------
.. [1] https://dlmf.nist.gov/10.47.E7
.. [2] https://dlmf.nist.gov/10.51.E5
.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
    Handbook of Mathematical Functions with Formulas,
    Graphs, and Mathematical Tables. New York: Dover, 1972.

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

>>> from scipy.special import spherical_in
>>> spherical_in(0, 3+5j)
(-1.1689867793369182-1.2697305267234222j)
>>> type(spherical_in(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_in(3, x, True),
...             spherical_in(2, x) - 4/x * spherical_in(3, x))
True

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

>>> import matplotlib.pyplot as plt
>>> x = np.arange(0.0, 6.0, 0.01)
>>> fig, ax = plt.subplots()
>>> ax.set_ylim(-0.5, 5.0)
>>> ax.set_title(r'Modified spherical Bessel functions $i_n$')
>>> for n in np.arange(0, 4):
...     ax.plot(x, spherical_in(n, x), label=rf'$i_{n}$')
>>> plt.legend(loc='best')
>>> plt.show()

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