As before, define the coefficients of and as functions of g and f .

For these of course reduce to much simpler expressions. The cos term becomes

This is identical to eq. (11) in TM97-01. For this expression is identically zero. The sin term becomes

Let's look at some plots.

Gplot := proc( F, vals::list, frange::range )
local p, k, Grange;
if nargs=3 then
Grange := args[3];
else
Grange := -2.5..2.5;
fi;
p := [];
for k from 1 to nops(gvals) do
p := [ op(p), plot( F(vals[k],f), f=frange,
color=mycolors[(k-1 mod 10)+1],
view=[frange,Grange], numpoints=200,
thickness=2 ) ];
od;
plots[display](p,axes=normal);
end:

Plot the cos term only:

These curves appear the same as those plotted in TM97-01. Hence, the function is probably identical to the G function in Appendix A of TM97-01. Now plot the sin term only:

Oh my. The sin term appears to have problems.

Animations for fun:

plots[animate]( G2[c](abs(t-1),f), f=-2.5..2.5, t=0..2, frames=25,
color=blue, axes=normal, thickness=2,
labels=["f",""], title="cosine term" );

plots[animate]( G2[s](abs(t-1),f), f=-2..2, t=0..2, frames=25,
color=blue, axes=normal, thickness=2,
view=[-2..2,-6..6],
labels=["f",""], title="sine term" );