On functions of finite analytical complexity

An analytic function of two complex variables has finite complexity, if it is representable as a finite superposition of summation and analytic functions of one variable. The minimal possible depth of the superposition is called analytical complexity of the function. To the best of the speaker’s knowledge, explicit examples of functions of complexity $n$, except for the cases $n=0,1,2$ and infinity, were not known before. We will give examples of analytic functions and also polynomials of an arbitrary given finite complexity.