PLegendreA_d1()

Compute all the unnormalized associated Legendre functions and first derivatives.

Usage

p, dp = PLegendreA_d1 (lmax, z, [csphase])

Returns

p : float, dimension ((lmax+1)*(lmax+2)/2)

An array of unnormalized 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 unnormalized 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.

Description

PLegendreA_d1 will calculate all of the unnormalized 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 hence will overflow for moderate values of l and m. 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 associated Legendre functions over the interval [-1, 1] is 2(l+m)!/(l-m)!/(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 polynomials 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.