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A generalization
of even and odd functions
Micki Balaich and Matthew Ondrus
Vol. 4 (2011), No. 1, 91–102
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Abstract |
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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. |
Keywords
complex function, group, representation
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Mathematical Subject Classification 2010
Primary: 30A99
Secondary: 20C15
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Milestones
Received: 14 September 2010
Revised: 2 May 2011
Accepted: 25 May 2011
Published: 22 September 2011
Communicated by Vadim Ponomarenko
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Authors |
| Micki Balaich | |
| Mathematics Department Weber State University 1702 University Circle Ogden, UT 84408 United States |
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| Matthew Ondrus | |
| Mathematics Department Weber State University 1702 University Circle Ogden, UT 84408 United States |
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