spharm_lm()

Compute the spherical harmonic function for specific degrees and orders.

Usage

ylm = spharm_lm (l, m, theta, phi, [normalization, kind, csphase, degrees])

Returns

ylm : float or complex, ndarray

The spherical harmonic function ylm, where l and m are the spherical harmonic degree and order, respectively.

Parameters

l : integer, array_like

The spherical harmonic degree.

m : integer, array_like

The spherical harmonic order.

theta : float, array_like

The colatitude in degrees. Use radians if ‘degrees’ is set to False.

phi : float, array_like

The longitude in degrees. Use radians if ‘degrees’ is set to False.

normalization : str, array_like, optional, default = ‘4pi’

‘4pi’, ‘ortho’, ‘schmidt’, or ‘unnorm’ for geodesy 4pi normalized, orthonormalized, Schmidt semi-normalized, or unnormalized spherical harmonic functions, respectively.

kind : str, array_like, optional, default = ‘real’

‘real’ or ‘complex’ spherical harmonic coefficients.

csphase : integer, array_like, optional, 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 spherical harmonic functions.

degrees : bool, array_like, optional, default = True

If True, theta and phi are expressed in degrees.

Notes

spharm_lm will calculate the spherical harmonic function for specific degrees l, orders m, colatitudes theta and longitudes phi. Three parameters determine how the spherical harmonic functions are defined. normalization can be either ‘4pi’ (default), ‘ortho’, ‘schmidt’, or ‘unnorm’ for 4pi normalized, orthonormalized, Schmidt semi-normalized, or unnormalized spherical harmonic functions, respectively. kind can be either ‘real’ or ‘complex’, and csphase determines whether to include or exclude (default) the Condon-Shortley phase factor.

The spherical harmonic functions are calculated using the standard three-term recursion formula, and in order to prevent overflows, the scaling approach of Holmes and Featherstone (2002) is utilized. The resulting functions are accurate to about degree 2800. See Wieczorek and Meschede (2018) for exact definitions on how the spherical harmonic functions are defined.

References

Holmes, S. A., and W. E. Featherstone, A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions, J. Geodesy, 76, 279-299, doi:10.1007/s00190-002-0216-2, 2002.

Wieczorek, M. A., and M. Meschede. SHTools — Tools for working with spherical harmonics, Geochem., Geophys., Geosyst., 19, 2574-2592, doi:10.1029/2018GC007529, 2018.