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| − | ''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p0721501.png" />'' | + | $f$ ''with domain'' $X$ |
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| − | A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p0721502.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p0721503.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p0721504.png" />) the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p0721505.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p0721506.png" /> also belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p0721507.png" /> and such that the following equality holds: | + | A number $T \ne 0$ such that for any $x \in X \subset \mathbf{R}$ (or $x \in X \subset \mathbf{C}$) the numbers $x+T$ and $x-T$ also belong to $X$ and such that the following equality holds: |
| | + | $$ |
| | + | f(x \pm T) = f(x) \ . |
| | + | $$ |
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| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p0721508.png" /></td> </tr></table>
| + | The numbers $\pm nT$, where $n$ is a natural number, are also periods of $f$. For a function $f=\text{const.}$ on an axis or on a plane, any number $T\ne0$ is a period; for the [[Dirichlet-function|Dirichlet function]] |
| − | | + | $$ |
| − | The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p0721509.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215010.png" /> is a natural number, are also periods of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215011.png" />. For a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215012.png" /> on an axis or on a plane, any number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215013.png" /> is a period; for the Dirichlet function | + | D(x) = \begin{cases} 1 &\text{if}\ x\ \text{is rational} \\ 0 & \text{if}\ x\ \text{is irrational} \end{cases} \ , |
| − | | + | $$ |
| − | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215014.png" /></td> </tr></table>
| + | any rational number $T\ne0$ is a period. If a function $f$ has period $T$, then the function $\psi(x) = f(ax+b)$, where $a$ and $b$ are constants and $a\ne0$, has period $T/a$. If a real-valued function $f$ of a real argument is continuous and periodic on $X$ (and is not identically equal to a constant), then it has a least period $T_0 > 0$ and any other real period is an integer multiple of $T_0$. There exist non-constant [[double-periodic function]]s of a complex argument, having two periods with non-real quotient, such as for example the [[elliptic function]]s. |
| − | | |
| − | any rational number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215015.png" /> is a period. If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215016.png" /> has period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215017.png" />, then the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215020.png" /> are constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215021.png" />, has period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215022.png" />. If a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215023.png" /> of a real argument is periodic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215024.png" /> (and is not identically equal to a constant), then it has a least period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215025.png" /> and any other real period is a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072150/p07215026.png" />. There exist non-constant functions of a complex argument having two non-multiple periods with imaginary quotient, such as for example an [[Elliptic function|elliptic function]]. | |
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| | Similarly one defines the period of a function defined on an Abelian group. | | Similarly one defines the period of a function defined on an Abelian group. |
| | | | |
| | + | ====Comments==== |
| | + | Cf. also [[Periodic function]]. |
| | | | |
| − | | + | {{TEX|done}} |
| − | ====Comments====
| |
| − | Cf. also [[Periodic function|Periodic function]].
| |
Latest revision as of 21:30, 18 November 2017
$f$ with domain $X$
A number $T \ne 0$ such that for any $x \in X \subset \mathbf{R}$ (or $x \in X \subset \mathbf{C}$) the numbers $x+T$ and $x-T$ also belong to $X$ and such that the following equality holds:
$$
f(x \pm T) = f(x) \ .
$$
The numbers $\pm nT$, where $n$ is a natural number, are also periods of $f$. For a function $f=\text{const.}$ on an axis or on a plane, any number $T\ne0$ is a period; for the Dirichlet function
$$
D(x) = \begin{cases} 1 &\text{if}\ x\ \text{is rational} \\ 0 & \text{if}\ x\ \text{is irrational} \end{cases} \ ,
$$
any rational number $T\ne0$ is a period. If a function $f$ has period $T$, then the function $\psi(x) = f(ax+b)$, where $a$ and $b$ are constants and $a\ne0$, has period $T/a$. If a real-valued function $f$ of a real argument is continuous and periodic on $X$ (and is not identically equal to a constant), then it has a least period $T_0 > 0$ and any other real period is an integer multiple of $T_0$. There exist non-constant double-periodic functions of a complex argument, having two periods with non-real quotient, such as for example the elliptic functions.
Similarly one defines the period of a function defined on an Abelian group.
Cf. also Periodic function.
How to Cite This Entry:
Period of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Period_of_a_function&oldid=13109
This article was adapted from an original article by A.A. Konyushkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article