scipy.special.clpmn¶

scipy.special.
clpmn
(m, n, z, type=3)[source]¶ 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)
containingPmn(z)
andPmn'(z)
for all orders from0..m
and degrees from0..n
. Parameters
 mint
m <= n
; the order of the Legendre function. nint
where
n >= 0
; the degree of the Legendre function. Often calledl
(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 axis1 < x < 1
(default)
 Returns
 Pmn_z(m+1, n+1) array
Values for all orders
0..m
and degrees0..n
 Pmn_d_z(m+1, n+1) array
Derivatives for all orders
0..m
and degrees0..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 atx > 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/f_src/special_functions/special_functions.html
 2
NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/14.21