plmschmidt_d1.doc

Compute all the Schmidt semi-normalized associated Legendre functions and first
derivatives.

Usage
-----
p, dp = PlmSchmidt_d1 (lmax, z, [csphase, cnorm])

Returns
-------
p : float, dimension ((lmax+1)*(lmax+2)/2)
    An array of Schmidt-normalized associated Legendre functions up to degree
    lmax. The index corresponds to l*(l+1)/2+m.
dp : float, dimension ((lmax+1)*(lmax+2)/2)
    An array of the first derivatives of the Schmidt-normalized associated
    Legendre functions up to degree lmax. The index corresponds to l*(l+1)/2+m.

Parameters
----------
lmax : integer
    The maximum degree of the associated Legendre functions to be computed.
z : float
    The argument of the associated Legendre functions.
csphase : optional, integer, default = 1
    If 1 (default), the Condon-Shortley phase will be excluded. If -1, the
    Condon-Shortley phase of (-1)^m will be appended to the associated Legendre
    functions.
cnorm : optional, integer, default = 0
    If 1, the complex normalization of the associated Legendre functions will be
    used. The default is to use the real normalization.

Description
-----------
PlmSchmidt_d1 will calculate all of the Schmidt-normalized associated Legendre
functions and first derivatives up to degree lmax for a given argument. These
are calculated using a standard three-term recursion formula, and in order to
prevent overflows, the scaling approach of Holmes and Featherstone (2002) is
utilized. These functions are accurate to about degree 2800. The index of the
array corresponding to a given degree l and angular order m corresponds to
l*(l+1)/2+m.

The integral of the squared Legendre functions over the interval [-1, 1] is
2*(2-delta(0,m))/(2l+1), where delta is the Kronecker delta function. If the
optional parameter cnorm is set equal to 1, the complex normalization will be
used where the integral of the squared Legendre functions over the interval [-1,
1] is 2/(2l+1). The default is to exclude the Condon-Shortley phase, but this
can be modified by setting the optional argument csphase to -1. Note that the
derivative of the Legendre functions is calculated with respect to its arguement
z, and not latitude or colatitude. If z=cos(theta), where theta is the
colatitude, then it is only necessary to multiply dp by -sin(theta) to obtain
the derivative with respect to theta.

References
----------
Holmes, S. A., and W. E. Featherstone, A unified approach to the Clenshaw
summation and the recursive computation of very high degree and
order normalised associated Legendre functions, J. Geodesy, 76, 279-
299, 2002.