quasiperiodic function QuasiperiodicFunction 2004-10-03 02:16:36 2008-02-23 19:38:11 Definition quasiperiod quasiperiodicity period periodic function periodic periodicity % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkescapeword{connection} \PMlinkescapeword{even} \PMlinkescapeword{meets} \PMlinkescapeword{order} \PMlinkescapeword{strictly} \PMlinkescapeword{term} 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.