SciPy

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

[R208]NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/14.3

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