Compute all the orthonormalized Legendre polynomials and first derivatives.
`p`, `dp` = PlON_d1 (`lmax`, `z`)
p
: float, dimension (lmax
+1)
: An array of orthonormalized Legendre polynomials up to degree lmax
. Degree l
corresponds to array index l
.
dp
: float, dimension (lmax
+1)
: An array of the first derivatives of the orthonormalized Legendre polynomials up to degree lmax
. Degree l
corresponds to array index l
.
lmax
: integer
: The maximum degree of the Legendre polynomials to be computed.
z
: float
: The argument of the Legendre polynomial.
PlON_d1
will calculate all of the orthonormalized Legendre polynomials and first derivatives up to degree lmax
for a given argument. These are calculated using a standard three-term recursion formula, and the integral of the orthonormalized Legendre polynomials over the interval [-1, 1] is 2/(4pi)
. 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
.