
Fundamental period for complex functions
Posted:
Aug 12, 2013 10:45 AM


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

