chaosmagpy.model_utils.legendre_poly

chaosmagpy.model_utils.legendre_poly(nmax, theta)[source]

Returns associated Legendre polynomials \(P_n^m(\cos\theta)\) (Schmidt quasi-normalized) and the derivative \(dP_n^m(\cos\theta)/d\theta\) evaluated at \(\theta\).

Parameters
nmaxint, positive

Maximum degree of the spherical expansion.

thetandarray, shape (…)

Colatitude in degrees \([0^\circ, 180^\circ]\) of arbitrary shape.

Returns
Pnmndarray, shape (n, m, …)

Evaluated values and derivatives, grid shape is appended as trailing dimensions. \(P_n^m(\cos\theta)\) := Pnm[n, m, ...] and \(dP_n^m(\cos\theta)/d\theta\) := Pnm[m, n+1, ...]

References

Based on Equations 26-29 and Table 2 in:

Langel, R. A., “Geomagnetism - The main field”, Academic Press, 1987, chapter 4