Let's make the coefficients of and into functions of ( g , f ).

Notice that the collection of ln terms in the sin term make for a very difficult time of keeping the sin term purely real. In fact, setting we have

where it becomes quite clear we're in trouble. There is no real value of f for which this expression does not have an imaginary component. On the other hand, the cos term is comparatively well-behaved:

Let's take a look at the cos term for various values of g .

cosplot := proc( gvals::list, frange::range )
local p, k;
p := [];
for k from 1 to nops(gvals) do
p := [ op(p), plot( G[c](gvals[k],f), f=frange,
color=mycolors[(k-1 mod 10)+1],
thickness=2 ) ];
od;
plots[display]( p, axes=normal );
end:

An animation of this is also instructive.

Hmm, disturbing that it is asymmetric.