Difficulties in Complex Multiplication and Exponentiation

TL;DR

This paper investigates the fundamental difficulties in complex exponentiation and multiplication, especially for non-integer rational powers greater than 2, using Riemann surfaces, and addresses questions about coding Riemann surface variables.

Contribution

It analyzes the challenges of defining complex exponentiation on Riemann surfaces and explores the incompatibility of addition with this structure.

Findings

Addition is incompatible with Riemann surface structure.

Complex exponentiation for non-integer powers faces fundamental difficulties.

Questions about coding Riemann surface variables are addressed.

Abstract

During my study of the iteration of functions of the form $f(z)=z^{\alpha}+c$, where $z,c \in \mathbbC$, and $\alpha$ is a rational non-integer larger than 2 (\cite{s1}), I encountered a fundamental difficulty in the exponentiation of a complex number. This paper will explore this difficulty and the problems encountered in trying to resolve it using a Riemann surface which is the direct generalization of the polar form of the complex plane. This paper will also answer two questions raised by Robert Corless in his \emph{E.C.C.A.D.} presentation \cite{co}: "Can a Riemann surface variable be coded? What will the operations be on it?" Unfortunately, the addition operation will be incompatible with the Riemann surface structure.