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# 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) 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, optional takes values 2 or 3 2: cut on the real axis |x| > 1 3: cut on the real axis -1 < x < 1 (default) 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 [R392] 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

 [R392] (1, 2) Zhang, Shanjie and Jin, Jianming. “Computation of Special Functions”, John Wiley and Sons, 1996. http://jin.ece.illinois.edu/specfunc.html
 [R393] NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/14.21