scipy.special.clpmn

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

Associated Legendre function of the first kind for complex arguments.

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

mint

|m| <= n; the order of the Legendre function.

nint

where n >= 0; the degree of the Legendre function. Often called l (lower case L) in descriptions of the associated Legendre function

zfloat or complex

Input value.

typeint, optional

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 [1] 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

1

Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. https://people.sc.fsu.edu/~jburkardt/f77_src/special_functions/special_functions.html

2

NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/14.21