'swsh_alpha_fits.dat' presents fitting parameters for the overlap
between two spheroidal harmonics, as a function of the spin
parameter j.
The functions we fit are
alpha_{m,l,l',n,n'}(j) = \int {}_{-2}S_{l,m,n} {}_{-2}S_{l',m,n'}(j)^* d\Omega
Because of symmetry, we present fits only for l < l', or l = l' and n < n'.
alpha_{m,l,l,n,n} is always unity, and we can recover the missing alpha's with
alpha_{m,l,l',n,n'} = alpha_{m,l',l,n',n}^*.
We present fitting functions as
Re(alpha_{m,l,l',n,n'}(j)) ~ delta_{l,l'} + p1r j^{p2r} + p3r j^{p4r}
and
Im(alpha_{m,l,l',n,n'}(j)) ~ p1i j^{p2i} + p3i j^{p4i}
with p2r and p2i in [0,20], p4r and p4i in [0, 100], p2r + 1 <= p4r, and p2i + 1 <= p4i
The columns listed are
m l l' n n' p1r p2r p3r p4r p1i p2i p3i p4i sr si mr mi
sr, si, mr, mi represent additional information on the fits:
- sr is the standard deviation between the real part of the fit and that of the
numerical function, defined as sr = sqrt((1/N) sum |Re(alpha_{fit} - alpha_{num})|^2)
- si is similar for the imaginary part,
si = sqrt((1/N) sum |Im(alpha_{fit} - alpha_{num})|^2)
- mr is the maximum absolute deviation between the real part of the fit and that
of the numerical function, mr = max_{N} |Re(alpha_{fit} - alpha_{num})|
- mi is similar for the imaginary part, mi = max_{N} |Im(alpha_{fit} - alpha_{num})|
The fits are least squares fits, i.e. p1r, ..., p4r minimize sr, and
p1i, ..., p4i minimize si.