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Compute Ricatti-Bessel function of the second kind and its derivative. The Ricatti-Bessel function of the second kind is defined as :math:`x y_n(x)`, where :math:`y_n` is the spherical Bessel function of the second kind of order :math:`n`. This function computes the value and first derivative of the function for all orders up to and including `n`. Parameters ---------- x : ndarray Argument at which to evaluate Returns ------- yn : ndarray Value of y0(x), ..., yn(x) ynp : ndarray First derivative y0'(x), ..., yn'(x) Notes ----- The computation is carried out via ascending recurrence, using the relation DLMF 10.51.1 [2]_. Wrapper for a Fortran routine created by Shanjie Zhang and Jianming Jin [1]_. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.51.E1 Compute Ricatti-Bessel function of the first kind and its derivative. The Ricatti-Bessel function of the first kind is defined as :math:`x j_n(x)`, where :math:`j_n` is the spherical Bessel function of the first kind of order :math:`n`. This function computes the value and first derivative of the Ricatti-Bessel function for all orders up to and including `n`. Parameters ---------- x : ndarray Argument at which to evaluate Returns ------- jn : ndarray Value of j0(x), ..., jn(x) jnp : ndarray First derivative j0'(x), ..., jn'(x) Notes ----- The computation is carried out via backward recurrence, using the relation DLMF 10.51.1 [2]_. Wrapper for a Fortran routine created by Shanjie Zhang and Jianming Jin [1]_. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/10.51.E1 Sequence of associated Legendre functions of the second kind. Computes the associated Legendre function of the second kind of order m and degree n, ``Qmn(z)`` = :math:`Q_n^m(z)`, and its derivative, ``Qmn'(z)``. Returns two arrays of size ``(m+1, n+1)`` containing ``Qmn(z)`` and ``Qmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. Parameters ---------- z : array_like Input value. Returns ------- Qmn_z : (m+1, n+1) array Values for all orders 0..m and degrees 0..n Qmn_d_z : (m+1, n+1) array Derivatives for all orders 0..m and degrees 0..n References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html Legendre function of the second kind. Compute sequence of Legendre functions of the second kind, Qn(z) and derivatives for all degrees from 0 to n (inclusive). References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html Associated Legendre function of the first kind for complex arguments. Computes the associated Legendre function of the first kind of order m and degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``. Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and ``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. Parameters ---------- z : array_like Input value. type : int takes values 2 or 3 2: cut on the real axis ``|x| > 1`` 3: cut on the real axis ``-1 < x < 1`` (default) m : array_like Sign of the order m. Returns ------- Pmn_z : (m+1, n+1) array Values for all orders ``0..m`` and degrees ``0..n`` Pmn_d_z : (m+1, n+1) array Derivatives for all orders ``0..m`` and degrees ``0..n`` See Also -------- lpmn: associated Legendre functions of the first kind for real z Notes ----- By default, i.e. for ``type=3``, phase conventions are chosen according to [1]_ such that the function is analytic. The cut lies on the interval (-1, 1). Approaching the cut from above or below in general yields a phase factor with respect to Ferrer's function of the first kind (cf. `lpmn`). For ``type=2`` a cut at ``|x| > 1`` is chosen. Approaching the real values on the interval (-1, 1) in the complex plane yields Ferrer's function of the first kind. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/14.21 Sequence of associated Legendre functions of the first kind. Computes the associated Legendre function of the first kind of order m and degree n, ``Pmn(z)`` = :math:`P_n^m(z)`, and its derivative, ``Pmn'(z)``. Returns two arrays of size ``(m+1, n+1)`` containing ``Pmn(z)`` and ``Pmn'(z)`` for all orders from ``0..m`` and degrees from ``0..n``. This function takes a real argument ``z``. For complex arguments ``z`` use clpmn instead. Parameters ---------- z : array_like Input value. m : array_like Sign of the order m. Returns ------- Pmn_z : (m+1, n+1) array Values for all orders 0..m and degrees 0..n Pmn_d_z : (m+1, n+1) array Derivatives for all orders 0..m and degrees 0..n See Also -------- clpmn: associated Legendre functions of the first kind for complex z Notes ----- In the interval (-1, 1), Ferrer's function of the first kind is returned. The phase convention used for the intervals (1, inf) and (-inf, -1) is such that the result is always real. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. "Computation of Special Functions", John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html .. [2] NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/14.3 Legendre function of the first kind. Compute sequence of Legendre functions of the first kind (polynomials), Pn(z) and derivatives for all degrees from 0 to n (inclusive). See also special.legendre for polynomial class. References ---------- .. [1] Zhang, Shanjie and Jin, Jianming. 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