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PLegendreA_d1
Compute all the unnormalized associated Legendre functions and first derivatives.
Usage
p
, dp
= pyshtools.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 tol*(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 tol*(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
.
See also
plbar
, lbar_d1
, plmbar
, plmbar_d1, plon, plon_d1
, plmon
, plmon_d1
, plschmidt
, plschmidt_d1
, plmschmidt
, plmschmidt_d1
, plegendre
, plegendre_d1
, plegendrea
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