Compute the spherical harmonic function for a specific degree and order.

Usage

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

Returns

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

Parameters

l : integer
The spherical harmonic degree.
m : integer
The spherical harmonic order.
theta : float
The colatitude in degrees.
phi : float
The longitude in degrees.
normalization : str, optional, default = ‘4pi’
‘4pi’, ‘ortho’, ‘schmidt’, or ‘unnorm’ for geodesy 4pi normalized, orthonormalized, Schmidt semi-normalized, or unnormalized spherical harmonic functions, respectively.
kind : str, optional, default = ‘real’
‘real’ or ‘complex’ spherical harmonic coefficients.
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 spherical harmonic functions.
degrees : optional, bool, default = True
If True, colat and phi are expressed in degrees.

Description

spharm_lm will calculate the spherical harmonic function for a specific degree l and order m, and for a given colatitude theta and longitude 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.

See also

pyspharm, pylegendre_lm, pylegendre

Tags: python
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