scipy.special.lpmn¶
- scipy.special.lpmn(m, n, z)[source]¶
Associated Legendre function of the first kind, Pmn(z).
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) 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: 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 : float
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
[R333] Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. http://jin.ece.illinois.edu/specfunc.html [R334] NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/14.3