Let F be a collection of holomorphic functions and let R(PR(F)) denote the reduct of the structure Ran to the ordered field operations together with the set of proper restrictions (see below) of the real and imaginary parts of all functions in F . We ask the question: Which holomorphic functions are locally definable (ie have their real and imaginary parts locally definable) in the structure R(PR(F))? It is easy to see that the collection of all such functions is closed under composition… CONTINUE READING