Difficulties in Complex Multiplication and Exponentiation

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.