scipy.special.lpmn#
- scipy.special.lpmn(m, n, z)[source]#
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)
= \(P_n^m(z)\), and its derivative,Pmn'(z)
. Returns two arrays of size(m+1, n+1)
containingPmn(z)
andPmn'(z)
for all orders from0..m
and degrees from0..n
.This function takes a real argument
z
. For complex argumentsz
use clpmn instead.- Parameters:
- mint
|m| <= n
; the order of the Legendre function.- nint
where
n >= 0
; the degree of the Legendre function. Often calledl
(lower case L) in descriptions of the associated Legendre function- zfloat
Input value.
- 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