Associativity and Distributivity of Analytic Functions

It is proved that, up to natural equivalence, there exist exactly five locally associative analytic functions of two variables: $0$, $x$, $y$, $(x+y)$, and $xy$. A description of all associative (nonlocally) polynomials is given. All distributive pairs where at least one of the functions is locally associative are also described.