Complex Iterations and Bounded Analytic Hyper-operators

Abstract

We give a method of solution to the problem of iterating holomorphic functions to fractional or complex heights. We construct an auxiliary function from natural iterates of a holomorphic function; the auxiliary function will be differintegrable and the complex derivatives of the auxiliary function are the complex iterates of the original holomorphic function. We use Ramanujan's master theorem as a foundation and apply elementary theorems from complex analysis to arrive at our result. We provide non-trivial examples of holomorphic functions iterated to complex heights using these methods. We derive a closed form expression for what we call bounded analytic hyper-operators. These hyper-operators share the same recursive structure as the hyper-operators defined on the natural numbers but are instead analytic. They form a sequence of operators beginning with addition, multiplication, and exponentiation. Surprisingly these hyper-operators are bounded by $e$ as they grow on the real line. We maintain an elementary yet very general discussion of the problem, as our solutions are specific instances of more general cases.