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/f_src/special_functions/special_functions.html
 2
NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/14.3