Function Reference
— Function File: l = legendre (n, x)
— Function File: l = legendre (n, x, normalization)

Compute the Legendre function of degree n and order m = 0 ... N. The optional argument, normalization, may be one of "unnorm", "sch", or "norm". The default is "unnorm". The value of n must be a non-negative scalar integer.

If the optional argument normalization is missing or is "unnorm", compute the Legendre function of degree n and order m and return all values for m = 0 ... n. The return value has one dimension more than x.

The Legendre Function of degree n and order m:

           m        m       2  m/2   d^m
          P(x) = (-1) * (1-x  )    * ----  P (x)
           n                         dx^m   n

with Legendre polynomial of degree n:

                    1     d^n   2    n
          P (x) = ------ [----(x - 1)  ]
           n      2^n n!  dx^n

legendre (3, [-1.0, -0.9, -0.8]) returns the matrix:

           x  |   -1.0   |   -0.9   |  -0.8
          ------------------------------------
          m=0 | -1.00000 | -0.47250 | -0.08000
          m=1 |  0.00000 | -1.99420 | -1.98000
          m=2 |  0.00000 | -2.56500 | -4.32000
          m=3 |  0.00000 | -1.24229 | -3.24000

If the optional argument normalization is "sch", compute the Schmidt semi-normalized associated Legendre function. The Schmidt semi-normalized associated Legendre function is related to the unnormalized Legendre functions by the following:

For Legendre functions of degree n and order 0:

            0       0
          SP (x) = P (x)
            n       n

For Legendre functions of degree n and order m:

            m       m          m    2(n-m)! 0.5
          SP (x) = P (x) * (-1)  * [-------]
            n       n               (n+m)!

If the optional argument normalization is "norm", compute the fully normalized associated Legendre function. The fully normalized associated Legendre function is related to the unnormalized Legendre functions by the following:

For Legendre functions of degree n and order m

            m       m          m    (n+0.5)(n-m)! 0.5
          NP (x) = P (x) * (-1)  * [-------------]
            n       n                   (n+m)!