The idea of using Leibniz rule also comes to mind.
sexp(x) = a0 + a1 x + a2 x^2/2 + ...
theta(x) = t0 + t1 x + r2 x^2/2 + ...
sexp2(x) = sexp(x+theta(x)) = b0 + b1 x + b2 x^2/2 + ...
sexp2 ' (x) = sexp ' (x+theta(x)) * (1+theta '(x))
set x = 1 and use Leibniz rule to make a system of equations.
also a0 = t0 = b0 = 0.
Get a proof by contradiction that is theta(x) is 1-periodic from the system of equations.
Needs more work ...
regards
tommy1729
sexp(x) = a0 + a1 x + a2 x^2/2 + ...
theta(x) = t0 + t1 x + r2 x^2/2 + ...
sexp2(x) = sexp(x+theta(x)) = b0 + b1 x + b2 x^2/2 + ...
sexp2 ' (x) = sexp ' (x+theta(x)) * (1+theta '(x))
set x = 1 and use Leibniz rule to make a system of equations.
also a0 = t0 = b0 = 0.
Get a proof by contradiction that is theta(x) is 1-periodic from the system of equations.
Needs more work ...
regards
tommy1729