PlanetMath: quasiperiodic function (original) (raw)

quasiperiodic function (Definition) A function $ f$ is said to have a quasiperiod $ p$ if there exists a function $ g$ such that $\displaystyle f(z + p) = g(z) f(z)$ In the special case where $ g$ is identially equal to $ 1$, we say that $ p$ is a period of $ f$ and that $ f$ is a 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. Also note that almost periodic functions are quite a different affair than quasiperiodic functions -- there, one is dealing with a precise notion. "quasiperiodic function" is owned by rspuzio. [ full author list (2) ] (view preamble) See Also: complex tangent and cotangent Also defines: quasiperiod, period, periodic function, periodic Cross-references: almost periodic functions, exponential function, complex number, gamma function, polynomial, rational function, functionThere are 27 references to this entry.This is version 8 of quasiperiodic function, born on 2004-10-03, modified 2006-10-05.Object id is 6280, canonical name is QuasiperiodicFunction.Accessed 6162 times total.Classification: AMS MSC: 30A99 (Functions of a complex variable :: General properties :: Miscellaneous)
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