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plmon.doc
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plmon.doc
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Compute all the orthonormalized associated Legendre functions.
Usage
-----
p = PlmON (lmax, z, [csphase, cnorm])
Returns
-------
p : float, dimension ((lmax+1)*(lmax+2)/2)
An array of orthonormalized 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
-----------
PlmON will calculate all of the orthonormalized associated Legendre functions 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 is l*(l+1)/2+m.
The integral of the squared Legendre functions over the interval [-1, 1] is
(2-delta(0,m))/(2pi), 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 1/(2pi). The default is to exclude the Condon-Shortley phase, but this can
be modified by setting the optional argument csphase to -1.
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.