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
n : int
z : float or complex
type : int
|
---|---|
Returns: | Pmn_z : (m+1, n+1) array
Pmn_d_z : (m+1, n+1) array
|
See also
Notes
By default, i.e. for type=3, phase conventions are chosen according to [R199] 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
[R199] | (1, 2) NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/14.21 |