A generalization of even and odd functions

Abstract

We generalize the concepts of even and odd functions in the setting of complex-valued functions of a complex variable. If $n>1$ is a fixed integer and $r$ is an integer with $0\le r<n$, we define what it means for a function to have type $rmodn$. When $n=2$, this reduces to the notions of even ($r=0$) and odd ($r=1$) functions respectively. We show that every function can be decomposed in a unique way as the sum of functions of types-0 through $n-1$. When the given function is differentiable, this decomposition is compatible with the differentiation operator in a natural way. We also show that under certain conditions, the type $r$ component of a given function may be regarded as a real-valued function of a real variable. Although this decomposition satisfies several analytic properties, the decomposition itself is largely algebraic, and we show that it can be explained in terms of representation theory.