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# scipy.special.clpmn

scipy.special.clpmn(m, n, z, type=3)

  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)

  Returns:
    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 [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 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

  [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