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'quasiperiodic function'
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| Title of object: |
quasiperiodic function |
| Canonical Name: |
QuasiperiodicFunction |
| Type: |
Definition |
| Created on: |
2004-10-03 02:16:36 |
| Modified on: |
2005-02-28 16:08:07 |
| Classification: |
msc:30A99 |
| Defines: |
quasiperiod, period, periodic function, periodic |
Revision comment (for changes between this and next version):
Preamble:
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Content:
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 identially equal to $1$, we say that $p$ is a \emph{period} of $f$ and that $f$ is a \emph{periodic} function.
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. |
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