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 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.
Cite
@article{arxiv.1503.07555,
title = {Complex Iterations and Bounded Analytic Hyper-operators},
author = {James Nixon},
journal= {arXiv preprint arXiv:1503.07555},
year = {2016}
}