I know $\exp(z) , \sin(z) , \sin^2(z) , z \exp(z) , \sinh(z) , \exp(z) \sin(z) $ all have addition formulas.
I'm still confused about addition formulas.
It seems hard to tell if a function has an addition formula or not.
Perhaps related : About the addition formula $f(x+y) = f(x)g(y)+f(y)g(x)$
Second question :
Does every function with an addition formula grow exponentially fast ?