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from __future__ import division, print_function, absolute_import
from ._ufuncs import (_spherical_jn, _spherical_yn, _spherical_in,
_spherical_kn, _spherical_jn_d, _spherical_yn_d,
_spherical_in_d, _spherical_kn_d)
def spherical_jn(n, z, derivative=False):
r"""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' = j_{n-1} - \frac{n + 1}{2} j_n.
j_0' = -j_1
.. versionadded:: 0.18.0
References
----------
.. [1] http://dlmf.nist.gov/10.47.E3
.. [2] http://dlmf.nist.gov/10.51.E1
.. [3] http://dlmf.nist.gov/10.51.E2
"""
if derivative:
return _spherical_jn_d(n, z)
else:
return _spherical_jn(n, z)
def spherical_yn(n, z, derivative=False):
r"""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}{2} y_n.
y_0' = -y_1
.. versionadded:: 0.18.0
References
----------
.. [1] http://dlmf.nist.gov/10.47.E4
.. [2] http://dlmf.nist.gov/10.51.E1
.. [3] http://dlmf.nist.gov/10.51.E2
"""
if derivative:
return _spherical_yn_d(n, z)
else:
return _spherical_yn(n, z)
def spherical_in(n, z, derivative=False):
r"""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}{2} i_n.
i_1' = i_0
.. versionadded:: 0.18.0
References
----------
.. [1] http://dlmf.nist.gov/10.47.E7
.. [2] http://dlmf.nist.gov/10.51.E5
"""
if derivative:
return _spherical_in_d(n, z)
else:
return _spherical_in(n, z)
def spherical_kn(n, z, derivative=False):
r"""Modified spherical Bessel function of the second kind or its derivative.
Defined as [1]_,
.. math:: k_n(z) = \sqrt{\frac{\pi}{2z}} K_{n + 1/2}(z),
where :math:`K_n` is the modified 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
-------
kn : ndarray
Notes
-----
The function is computed using its definitional relation to the
modified cylindrical Bessel function of the second kind.
The derivative is computed using the relations [2]_,
.. math::
k_n' = -k_{n-1} - \frac{n + 1}{2} k_n.
k_0' = -k_1
.. versionadded:: 0.18.0
References
----------
.. [1] http://dlmf.nist.gov/10.47.E9
.. [2] http://dlmf.nist.gov/10.51.E5
"""
if derivative:
return _spherical_kn_d(n, z)
else:
return _spherical_kn(n, z)
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