Abstract
We generalize the concepts of even and odd functions in the setting of complex-valued functions of a complex variable. If is a fixed integer and is an integer with , we define what it means for a function to have type . When , this reduces to the notions of even () and odd () functions respectively. We show that every function can be decomposed in a unique way as the sum of functions of types-0 through . 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 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.
Citation
Micki Balaich. Matthew Ondrus. "A generalization of even and odd functions." Involve 4 (1) 91 - 102, 2011. https://doi.org/10.2140/involve.2011.4.91
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