Multiply two functions and determine the spherical harmonic coefficients of the resulting function.

## Usage

call SHMultiply (shout, sh1, lmax1, sh2, lmax2, precomp, norm, csphase, exitstatus)

## Parameters

shout : output, real(dp), dimension (2, lmax1+lmax2+1, lmax1+lmax2+1)
The real spherical harmonic coefficients corresponding to the multiplication of sh1 and sh2 in the space domain.
sh1 : input, real(dp), dimension (2, lmax1+1, lmax1+1)
The spherical harmonic coefficients of the first function.
lmax1 : input, integer
The maximum spherical harmonic degree used in evaluting sh1.
sh2 : input, real(dp), dimension (2, lmax2+1, lmax2+1)
The spherical harmonic coefficients of the second function.
lmax2 : input, integer
The maximum spherical harmonic degree used in evaluting sh2.
precomp : input, optional, integer, default = 0
If 1, the array of Legendre functions plx will be precomputed on the Gauss-Legendre quadrature nodes.
norm : input, optional, integer, default = 1
1 (default) = Geodesy 4-pi normalized harmonics; 2 = Schmidt semi-normalized harmonics; 3 = unnormalized harmonics; 4 = orthonormal harmonics.
csphase : input, optional, integer, default = 1
1 (default) = do not apply the Condon-Shortley phase factor to the associated Legendre functions; -1 = append the Condon-Shortley phase factor of (-1)^m to the associated Legendre functions.
exitstatus : output, optional, integer
If present, instead of executing a STOP when an error is encountered, the variable exitstatus will be returned describing the error. 0 = No errors; 1 = Improper dimensions of input array; 2 = Improper bounds for input variable; 3 = Error allocating memory; 4 = File IO error.

## Description

SHMultiply will take two sets of spherical harmonic coefficients, multiply the functions in the space domain, and expand the resulting field in spherical harmonics using SHExpandGLQ. The spherical harmonic bandwidth of the resulting field is lmax1+lmax2, where lmax1 and lmax2 are the bandwidths of the input fields. If the optional parameter precomp is set, then the array of Legendre functions plx will be precomputed on the Gauss-Legendre quadrature nodes.

The employed spherical harmonic normalization and Condon-Shortley phase convention can be set by the optional arguments norm and csphase; if not set, the default is to use geodesy 4-pi normalized harmonics that exclude the Condon-Shortley phase of (-1)^m.