lpmn(m, n, z)¶
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
Pmn'(z)for all orders from
0..mand degrees from
This function takes a real argument
z. For complex arguments
zuse clpmn instead.
|m| <= n; the order of the Legendre function.
n >= 0; the degree of the Legendre function. Often called
l(lower case L) in descriptions of the associated Legendre function
- 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
associated Legendre functions of the first kind for complex z
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.
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
NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/14.3