I was thinking about fourier series. Alot is known about it.
But I wonder
$$\exp(a_0 + a_1 \sin(x) + a_2 \sin(2x) + a_3 \sin(3x) + ...) = b_0 + b_1 \sin(x) + b_2 \sin(2x)+ ... + c_1 \cos(x) + c_2 \cos(2x) + ...$$
Is it possible to "simply" express $a_i$ in terms of $b_i,c_i$ or vice versa ?
I use "simply" because ofcourse we can use the fourier series coefficients formula for the LHS and set then equal to the coefficients formula of the RHS. That would give a system of equations such as $\int f(x) \sin(nx) dx = \int \exp(f(y)) \sin(ny) dy $. OR we could in the analytic case take the taylor series on both sides and equate them. OR ( also in the analytic case) we could use carleman matrices for function composition. But I mean something " simple " like a system of lineair equations or polynomial relationships.