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(1,2)
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