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Module « scipy.special »

Fonction clpmn - module scipy.special

Signature de la fonction clpmn

def clpmn(m, n, z, type=3) 

Description

help(scipy.special.clpmn)

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``.

.. deprecated:: 1.15.0
    This function is deprecated and will be removed in SciPy 1.17.0.
    Please use `scipy.special.assoc_legendre_p_all` instead.

Parameters
----------
m : int
   ``|m| <= n``; the order of the Legendre function.
n : int
   where ``n >= 0``; the degree of the Legendre function.  Often
   called ``l`` (lower case L) in descriptions of the associated
   Legendre function
z : array_like, float or complex
    Input value.
type : int, optional
   takes values 2 or 3
   2: cut on the real axis ``|x| > 1``
   3: cut on the real axis ``-1 < x < 1`` (default)

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

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