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. 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

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

 [R334] Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. http://jin.ece.illinois.edu/specfunc.html
 [R335] NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/14.3

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