I am familiar with the trig identitysum{n=0->N} sin(nx) = {sin(Nx/2).sin((N+1)x/2)}/sin(x/2)
that is,
derived via exponential sums (see https://mathworld.wolfram.com/Sine.html formulas (11) to (15)).
In my research (geophysics) I could really make use of a similar simplification if one can be derived, for the slightly more complicated sine series
sum{n=0->N} p_n.sin(q.n.x+r_n)
where p_n is a set of N+1 real numbers
q is a real number
r_n is a set of N+1 real numbers
that is,
I do not know if this is possible. If it is, it may bring a new technique into the realms of being computationally achievable! We shall see. Simplifications of the above kind, or any alternate ideas would be gratefully received, or even qualified statements as to why this isnt going to be simplified would be helpful.
Thanks,
Ben.