205 lines
5.1 KiB
Python
205 lines
5.1 KiB
Python
from __future__ import division, print_function, absolute_import
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from ._ufuncs import (_spherical_jn, _spherical_yn, _spherical_in,
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_spherical_kn, _spherical_jn_d, _spherical_yn_d,
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_spherical_in_d, _spherical_kn_d)
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def spherical_jn(n, z, derivative=False):
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r"""Spherical Bessel function of the first kind or its derivative.
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Defined as [1]_,
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.. math:: j_n(z) = \sqrt{\frac{\pi}{2z}} J_{n + 1/2}(z),
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where :math:`J_n` is the Bessel function of the first kind.
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Parameters
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----------
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n : int, array_like
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Order of the Bessel function (n >= 0).
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z : complex or float, array_like
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Argument of the Bessel function.
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derivative : bool, optional
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If True, the value of the derivative (rather than the function
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itself) is returned.
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Returns
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-------
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jn : ndarray
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Notes
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-----
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For real arguments greater than the order, the function is computed
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using the ascending recurrence [2]_. For small real or complex
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arguments, the definitional relation to the cylindrical Bessel function
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of the first kind is used.
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The derivative is computed using the relations [3]_,
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.. math::
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j_n'(z) = j_{n-1}(z) - \frac{n + 1}{z} j_n(z).
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j_0'(z) = -j_1(z)
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.. versionadded:: 0.18.0
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References
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----------
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.. [1] http://dlmf.nist.gov/10.47.E3
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.. [2] http://dlmf.nist.gov/10.51.E1
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.. [3] http://dlmf.nist.gov/10.51.E2
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"""
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if derivative:
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return _spherical_jn_d(n, z)
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else:
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return _spherical_jn(n, z)
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def spherical_yn(n, z, derivative=False):
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r"""Spherical Bessel function of the second kind or its derivative.
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Defined as [1]_,
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.. math:: y_n(z) = \sqrt{\frac{\pi}{2z}} Y_{n + 1/2}(z),
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where :math:`Y_n` is the Bessel function of the second kind.
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Parameters
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----------
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n : int, array_like
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Order of the Bessel function (n >= 0).
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z : complex or float, array_like
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Argument of the Bessel function.
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derivative : bool, optional
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If True, the value of the derivative (rather than the function
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itself) is returned.
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Returns
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-------
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yn : ndarray
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Notes
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-----
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For real arguments, the function is computed using the ascending
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recurrence [2]_. For complex arguments, the definitional relation to
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the cylindrical Bessel function of the second kind is used.
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The derivative is computed using the relations [3]_,
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.. math::
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y_n' = y_{n-1} - \frac{n + 1}{z} y_n.
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y_0' = -y_1
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.. versionadded:: 0.18.0
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References
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----------
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.. [1] http://dlmf.nist.gov/10.47.E4
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.. [2] http://dlmf.nist.gov/10.51.E1
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.. [3] http://dlmf.nist.gov/10.51.E2
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"""
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if derivative:
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return _spherical_yn_d(n, z)
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else:
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return _spherical_yn(n, z)
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def spherical_in(n, z, derivative=False):
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r"""Modified spherical Bessel function of the first kind or its derivative.
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Defined as [1]_,
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.. math:: i_n(z) = \sqrt{\frac{\pi}{2z}} I_{n + 1/2}(z),
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where :math:`I_n` is the modified Bessel function of the first kind.
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Parameters
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----------
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n : int, array_like
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Order of the Bessel function (n >= 0).
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z : complex or float, array_like
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Argument of the Bessel function.
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derivative : bool, optional
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If True, the value of the derivative (rather than the function
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itself) is returned.
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Returns
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-------
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in : ndarray
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Notes
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-----
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The function is computed using its definitional relation to the
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modified cylindrical Bessel function of the first kind.
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The derivative is computed using the relations [2]_,
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.. math::
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i_n' = i_{n-1} - \frac{n + 1}{z} i_n.
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i_1' = i_0
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.. versionadded:: 0.18.0
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References
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----------
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.. [1] http://dlmf.nist.gov/10.47.E7
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.. [2] http://dlmf.nist.gov/10.51.E5
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"""
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if derivative:
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return _spherical_in_d(n, z)
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else:
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return _spherical_in(n, z)
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def spherical_kn(n, z, derivative=False):
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r"""Modified spherical Bessel function of the second kind or its derivative.
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Defined as [1]_,
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.. math:: k_n(z) = \sqrt{\frac{\pi}{2z}} K_{n + 1/2}(z),
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where :math:`K_n` is the modified Bessel function of the second kind.
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Parameters
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----------
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n : int, array_like
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Order of the Bessel function (n >= 0).
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z : complex or float, array_like
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Argument of the Bessel function.
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derivative : bool, optional
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If True, the value of the derivative (rather than the function
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itself) is returned.
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Returns
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-------
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kn : ndarray
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Notes
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-----
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The function is computed using its definitional relation to the
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modified cylindrical Bessel function of the second kind.
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The derivative is computed using the relations [2]_,
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.. math::
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k_n' = -k_{n-1} - \frac{n + 1}{z} k_n.
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k_0' = -k_1
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.. versionadded:: 0.18.0
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References
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----------
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.. [1] http://dlmf.nist.gov/10.47.E9
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.. [2] http://dlmf.nist.gov/10.51.E5
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"""
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if derivative:
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return _spherical_kn_d(n, z)
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else:
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return _spherical_kn(n, z)
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