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PlSchmidt_d1() |
spherical harmonics software package, spherical harmonic transform, legendre functions, multitaper spectral analysis, Python, gravity, magnetic field |
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pyplschmidt_d1.html |
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pydoc |
Compute all the Schmidt-normalized Legendre polynomials and first derivatives.
p, dp = PlSchmidt_d1 (lmax, z)
p : float, dimension (lmax+1) : An array of Schmidt-normalized 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 Schmidt-normalized 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.
PlSchmidt_d1 will calculate all of the Schmidt-normalized 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 Schmidt-normalized Legendre polynomials over the interval [-1, 1] is 2/(2l+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.