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Topic: Fundamental period for complex functions
Replies: 3   Last Post: Aug 16, 2013 10:11 PM

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Posts: 17
Registered: 8/18/12
Fundamental period for complex functions
Posted: Aug 12, 2013 10:45 AM
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Suppose f from the complexes to the complexes is periodic. Then, is there something like a fundamental period for f?

On the reals, if f is periodic then f may have a fundamental period, defined, when it exists, as the smallest period of f. By definition, periods are positive, and, if P is the set of periods of f, then the fundamental period is p*= infimum P, if p* > 0. In this case, we can show p* is in P. So, in this case, p* is the minimum of P. And it's known that if f is continuous, periodic and non constant, then f has a fundamental period (aka minimum period).

In the complexes, the above definition doesn't make sense, but maybe we can define p* as the period with minimum absolute value, provided it's positive. So, the definition might be p = minimum {|p| : p is period of f}. Does this exist?

For example, 2?i is a period of f(z) = e^z. Does f have a period with positive absolute value < 2??

If we define p* for the complexes in this way, then, supposing f is continuous, periodic and non constant, then does such p* exist?

Thank you

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