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_spherical_kn, _spherical_jn_d, _spherical_yn_d, _spherical_in_d, _spherical_kn_d)
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'(z) = j_{n-1}(z) - \frac{n + 1}{z} j_n(z).
j_0'(z) = -j_1(z)
.. 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)
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}{z} 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)
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}{z} 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)
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}{z} 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) |