355 lines
10 KiB
Python
355 lines
10 KiB
Python
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import numpy as np
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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] https://dlmf.nist.gov/10.47.E3
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.. [2] https://dlmf.nist.gov/10.51.E1
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.. [3] https://dlmf.nist.gov/10.51.E2
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.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
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Handbook of Mathematical Functions with Formulas,
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Graphs, and Mathematical Tables. New York: Dover, 1972.
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Examples
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--------
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The spherical Bessel functions of the first kind :math:`j_n` accept
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both real and complex second argument. They can return a complex type:
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>>> from scipy.special import spherical_jn
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>>> spherical_jn(0, 3+5j)
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(-9.878987731663194-8.021894345786002j)
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>>> type(spherical_jn(0, 3+5j))
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<class 'numpy.complex128'>
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We can verify the relation for the derivative from the Notes
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for :math:`n=3` in the interval :math:`[1, 2]`:
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>>> import numpy as np
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>>> x = np.arange(1.0, 2.0, 0.01)
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>>> np.allclose(spherical_jn(3, x, True),
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... spherical_jn(2, x) - 4/x * spherical_jn(3, x))
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True
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The first few :math:`j_n` with real argument:
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>>> import matplotlib.pyplot as plt
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>>> x = np.arange(0.0, 10.0, 0.01)
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>>> fig, ax = plt.subplots()
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>>> ax.set_ylim(-0.5, 1.5)
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>>> ax.set_title(r'Spherical Bessel functions $j_n$')
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>>> for n in np.arange(0, 4):
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... ax.plot(x, spherical_jn(n, x), label=rf'$j_{n}$')
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>>> plt.legend(loc='best')
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>>> plt.show()
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"""
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n = np.asarray(n, dtype=np.dtype("long"))
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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] https://dlmf.nist.gov/10.47.E4
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.. [2] https://dlmf.nist.gov/10.51.E1
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.. [3] https://dlmf.nist.gov/10.51.E2
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.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
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Handbook of Mathematical Functions with Formulas,
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Graphs, and Mathematical Tables. New York: Dover, 1972.
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Examples
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--------
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The spherical Bessel functions of the second kind :math:`y_n` accept
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both real and complex second argument. They can return a complex type:
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>>> from scipy.special import spherical_yn
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>>> spherical_yn(0, 3+5j)
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(8.022343088587197-9.880052589376795j)
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>>> type(spherical_yn(0, 3+5j))
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<class 'numpy.complex128'>
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We can verify the relation for the derivative from the Notes
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for :math:`n=3` in the interval :math:`[1, 2]`:
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>>> import numpy as np
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>>> x = np.arange(1.0, 2.0, 0.01)
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>>> np.allclose(spherical_yn(3, x, True),
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... spherical_yn(2, x) - 4/x * spherical_yn(3, x))
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True
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The first few :math:`y_n` with real argument:
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>>> import matplotlib.pyplot as plt
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>>> x = np.arange(0.0, 10.0, 0.01)
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>>> fig, ax = plt.subplots()
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>>> ax.set_ylim(-2.0, 1.0)
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>>> ax.set_title(r'Spherical Bessel functions $y_n$')
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>>> for n in np.arange(0, 4):
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... ax.plot(x, spherical_yn(n, x), label=rf'$y_{n}$')
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>>> plt.legend(loc='best')
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>>> plt.show()
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"""
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n = np.asarray(n, dtype=np.dtype("long"))
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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] https://dlmf.nist.gov/10.47.E7
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.. [2] https://dlmf.nist.gov/10.51.E5
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.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
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Handbook of Mathematical Functions with Formulas,
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Graphs, and Mathematical Tables. New York: Dover, 1972.
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Examples
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--------
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The modified spherical Bessel functions of the first kind :math:`i_n`
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accept both real and complex second argument.
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They can return a complex type:
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>>> from scipy.special import spherical_in
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>>> spherical_in(0, 3+5j)
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(-1.1689867793369182-1.2697305267234222j)
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>>> type(spherical_in(0, 3+5j))
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<class 'numpy.complex128'>
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We can verify the relation for the derivative from the Notes
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for :math:`n=3` in the interval :math:`[1, 2]`:
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>>> import numpy as np
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>>> x = np.arange(1.0, 2.0, 0.01)
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>>> np.allclose(spherical_in(3, x, True),
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... spherical_in(2, x) - 4/x * spherical_in(3, x))
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True
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The first few :math:`i_n` with real argument:
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>>> import matplotlib.pyplot as plt
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>>> x = np.arange(0.0, 6.0, 0.01)
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>>> fig, ax = plt.subplots()
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>>> ax.set_ylim(-0.5, 5.0)
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>>> ax.set_title(r'Modified spherical Bessel functions $i_n$')
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>>> for n in np.arange(0, 4):
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... ax.plot(x, spherical_in(n, x), label=rf'$i_{n}$')
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>>> plt.legend(loc='best')
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>>> plt.show()
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"""
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n = np.asarray(n, dtype=np.dtype("long"))
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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] https://dlmf.nist.gov/10.47.E9
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.. [2] https://dlmf.nist.gov/10.51.E5
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.. [AS] Milton Abramowitz and Irene A. Stegun, eds.
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Handbook of Mathematical Functions with Formulas,
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Graphs, and Mathematical Tables. New York: Dover, 1972.
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Examples
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--------
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The modified spherical Bessel functions of the second kind :math:`k_n`
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accept both real and complex second argument.
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They can return a complex type:
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>>> from scipy.special import spherical_kn
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>>> spherical_kn(0, 3+5j)
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(0.012985785614001561+0.003354691603137546j)
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>>> type(spherical_kn(0, 3+5j))
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<class 'numpy.complex128'>
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We can verify the relation for the derivative from the Notes
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for :math:`n=3` in the interval :math:`[1, 2]`:
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>>> import numpy as np
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>>> x = np.arange(1.0, 2.0, 0.01)
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>>> np.allclose(spherical_kn(3, x, True),
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... - 4/x * spherical_kn(3, x) - spherical_kn(2, x))
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True
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The first few :math:`k_n` with real argument:
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>>> import matplotlib.pyplot as plt
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>>> x = np.arange(0.0, 4.0, 0.01)
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>>> fig, ax = plt.subplots()
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>>> ax.set_ylim(0.0, 5.0)
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>>> ax.set_title(r'Modified spherical Bessel functions $k_n$')
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>>> for n in np.arange(0, 4):
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... ax.plot(x, spherical_kn(n, x), label=rf'$k_{n}$')
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>>> plt.legend(loc='best')
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>>> plt.show()
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"""
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n = np.asarray(n, dtype=np.dtype("long"))
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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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