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Difference between revisions of "Simply-periodic function"

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''simple periodic function''
 
''simple periodic function''
  
A [[Periodic function|periodic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s0854401.png" /> of the complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s0854402.png" /> all periods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s0854403.png" /> of which are integer multiples of a single unique fundamental, or primitive, period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s0854404.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s0854405.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s0854406.png" />). For example, the [[Exponential function|exponential function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s0854407.png" /> is an entire simply-periodic function with fundamental period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s0854408.png" />, and the [[Trigonometric functions|trigonometric functions]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s0854409.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s08544010.png" /> are meromorphic simply-periodic functions with fundamental period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s08544011.png" />.
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A [[Periodic function|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|exponential function]] $e^z$ is an entire simply-periodic function with fundamental period $2\omega=2\pi i$, and the [[Trigonometric functions|trigonometric functions]] $\tan z$ and $\operatorname{cotan}z$ are meromorphic simply-periodic functions with fundamental period $2\omega=\pi$.
  
  
  
 
====Comments====
 
====Comments====
More generally, a simply-periodic function on a linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s08544012.png" /> is a periodic function whose periods are integer multiples of some basic period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085440/s08544013.png" />. A non-constant continuous periodic function of a real variable is necessarily simply-periodic.
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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.

Revision as of 08:14, 27 April 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.

How to Cite This Entry:
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