The kernels of the Orbital FTs as defined in Equation 
may be calculated analytically as an integral over the angular polar
coordinates.
The s spherical harmonic 
which has no angular dependence
gives rise to the simplest kernel
Higher order spherical harmonics give rise to more complex kernels,
for example 
gives
Here 
, and similarly we define 
and 
.
The kernels of spherical harmonics with non-zero m must be
constructed by finding the kernels corresponding to linear
combinations of spherical harmonics with equal l and m of equal
magnitude, but opposite sign. These combinations are constructed such
that the expressions generated are either purely imaginary or
real. The kernels of these linear combinations may then be combined to
generate the kernels corresponding to the original spherical harmonic.
For example the 
and 
kernels may be found as
follows:
The kernels 
and 
may be found in the same way as
above and are:
Therefore and are given by
The kernels for the d orbitals may be calculated in a similar way and are;
where 
, 
, etc.