30A99-QuasiperiodicFunction.tex

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\pmtitle{quasiperiodic function}
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\pmtype{Definition}
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\pmclassification{msc}{30A99}
\pmrelated{ComplexTangentAndCotangent}
\pmrelated{CounterperiodicFunction}
\pmdefines{quasiperiod}
\pmdefines{quasiperiodicity}
\pmdefines{period}
\pmdefines{periodic function}
\pmdefines{periodic}
\pmdefines{periodicity}

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A function $f$ is said to have a \emph{quasiperiod} $p$ if there exists a function $g$ such that
 $$f(z + p) = g(z) f(z).$$

In the special case where $g$ is identically equal to $1$, we call $f$ a \emph{periodic function}, and we say that $p$ is a \emph{period} of $f$ or that $f$ has \emph{periodicity} $p$.

Except for the special case of periodicity noted above, the notion of quasiperiodicity is somewhat loose and fuzzy.  Strictly speaking, many functions could be regarded as quasiperiodic if one defines $g(z) = f(z+p) / f(z)$.  In order for the term ``quasiperiodic'' not to be trivial, it is customary to reserve its use for the case where the function $g$ is, in some vague, intuitive sense, simpler than the function $f$.  For instance, no one would call the function $f(z) = z^2 + 1$ quasiperiodic even though it meets the criterion of the definition if we set $g(z) = (z^2 + 2z + 2) / (z^2 + 1)$ because the rational function $g$ is ``more complicated'' than the polynomial $f$.  On the other hand, for the gamma function, one would say that $1$ is a quasiperiod because $\Gamma (z+1) = z \Gamma(z)$ and the function $g(z) = z$ is a ``much simpler'' function than the gamma function.

Note that the every complex number can be said to be a quasiperiod of the exponential function.  The term ``quasiperiod'' is most frequently used in connection with theta functions.

Also note that almost periodic functions are quite a different affair than quasiperiodic functions --- there, one is dealing with a precise notion.
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