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..mand degrees from- 0..n.- Deprecated since version 1.15.0: This function is deprecated and will be removed in SciPy 1.17.0. Please use - scipy.special.assoc_legendre_p_allinstead.- 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
- zarray_like, float or complex
- Input value. 
- typeint, optional
- takes values 2 or 3 2: cut on the real axis - |x| > 13: cut on the real axis- -1 < x < 1(default)
 
- Returns:
- Pmn_z(m+1, n+1) array
- Values for all orders - 0..mand degrees- 0..n
- Pmn_d_z(m+1, n+1) array
- Derivatives for all orders - 0..mand 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=2a cut at- |x| > 1is 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