Vol. 4, No. 1, 2011

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A generalization of even and odd functions

Micki Balaich and Matthew Ondrus

Vol. 4 (2011), No. 1, 91–102

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 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.

complex function, group, representation
Mathematical Subject Classification 2010
Primary: 30A99
Secondary: 20C15
Received: 14 September 2010
Revised: 2 May 2011
Accepted: 25 May 2011
Published: 22 September 2011

Communicated by Vadim Ponomarenko
Micki Balaich
Mathematics Department
Weber State University
1702 University Circle
Ogden, UT 84408
United States
Matthew Ondrus
Mathematics Department
Weber State University
1702 University Circle
Ogden, UT 84408
United States