SciPy

scipy.special.clpmn

scipy.special.clpmn(m, n, z, type=3)[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.

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 or complex

Input value.

type : int

takes values 2 or 3 2: cut on the real axis |x|>1 3: cut on the real axis -1<x<1 (default)

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

lpmn
associated Legendre functions of the first kind for real z

Notes

By default, i.e. for type=3, phase conventions are chosen according to [R197] such that the function is analytic. The cut lies on the interval (-1, 1). Approaching the cut from above or below in general yields a phase factor with respect to Ferrer’s function of the first kind (cf. lpmn).

For type=2 a cut at |x|>1 is chosen. Approaching the real values on the interval (-1, 1) in the complex plane yields Ferrer’s function of the first kind.

References

[R197](1, 2) NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/14.21

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