Difference between revisions of "Simply-periodic function"
From Encyclopedia of Mathematics
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| − | A [[ | + | A [[periodic function]] $f(z)$ of the complex variable $z$ all periods $p$ of which are integer multiples of a single unique fundamental, or primitive, period $2\omega\neq0$, i.e. $p=2n\omega$ ($n\in\mathbf Z$). For example, the [[exponential function]] $e^z$ is an entire simply-periodic function with fundamental period $2\omega=2\pi i$, and the [[trigonometric functions]] $\tan z$ and $\operatorname{cotan}z$ are meromorphic simply-periodic functions with fundamental period $2\omega=\pi$. |
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| − | More generally, a simply-periodic function on a linear space | + | More generally, a simply-periodic function on a linear space $X$ is a periodic function whose periods are integer multiples of some basic period $2\omega\in X$. A non-constant continuous periodic function of a real variable is necessarily simply-periodic. |
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| + | See also [[Double-periodic function]] | ||
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| + | [[Category:Functions of a complex variable]] | ||
Latest revision as of 18:55, 17 October 2014
simple periodic function
A periodic function $f(z)$ of the complex variable $z$ all periods $p$ of which are integer multiples of a single unique fundamental, or primitive, period $2\omega\neq0$, i.e. $p=2n\omega$ ($n\in\mathbf Z$). For example, the exponential function $e^z$ is an entire simply-periodic function with fundamental period $2\omega=2\pi i$, and the trigonometric functions $\tan z$ and $\operatorname{cotan}z$ are meromorphic simply-periodic functions with fundamental period $2\omega=\pi$.
Comments
More generally, a simply-periodic function on a linear space $X$ is a periodic function whose periods are integer multiples of some basic period $2\omega\in X$. A non-constant continuous periodic function of a real variable is necessarily simply-periodic.
See also Double-periodic function
How to Cite This Entry:
Simply-periodic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simply-periodic_function&oldid=11772
Simply-periodic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Simply-periodic_function&oldid=11772
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article