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{{Short description|Function that repeats its values in regular intervals or periods}}
{{Distinguish|periodic mapping}}
{{Redirect-distinguish|Period length|repeating decimal}}
{{Redirect2|Aperiodic|Non-periodic}}

In [[mathematics]],  a '''periodic function'''  is a [[function (mathematics)|function]] that repeats its values in regular intervals or periods.  The most important examples are the [[trigonometric functions]], which repeat over intervals of 2π [[radian]]s. Periodic functions are used throughout science to describe [[oscillation]]s, [[wave]]s, and other phenomena that exhibit [[Frequency|periodicity]]. Any function that is not periodic is called '''aperiodic'''.

[[Image:Periodic function illustration.svg|thumb|right|300px|An illustration of a periodic function with period <math>P.</math>]]

==Definition==
A function {{math|<var>f</var>}} is said to be '''periodic''' if, for some '''nonzero''' constant {{math|<var>P</var>}}, it is the case that

:<math>f(x+P) = f(x) </math>

for all values of {{math|<var>x</var>}} in the domain. A nonzero constant {{mvar|P}} for which this is the case is called a '''period''' of the function. If there exists a least positive<ref>For some functions, like a [[constant function]] or the [[indicator function]] of the [[rational number]]s, a least positive period may not exist (the [[infimum]] of all positive periods {{math|<var>P</var>}} being zero).</ref> constant {{math|<var>P</var>}} with this property, it is called the '''fundamental period''' (also '''primitive period''', '''basic period''', or '''prime period'''.)  Often, "the" period of a function is used to mean its fundamental period. A function with period {{math|<var>P</var>}} will repeat on intervals of length {{math|<var>P</var>}}, and these intervals are sometimes also referred to as '''periods''' of the function.

Geometrically, a periodic function can be defined as a function whose graph exhibits [[translational symmetry]].  Specifically, a function {{math|<var>f</var>}} is periodic with period {{math|<var>P</var>}} if the graph of {{math|<var>f</var>}} is [[invariant (mathematics)|invariant]] under [[translation (geometry)|translation]] in the {{math|<var>x</var>}}-direction by a distance of {{math|<var>P</var>}}.  This definition of periodic can be extended to other geometric shapes and patterns, as well as be generalized to higher dimensions, such as periodic [[tessellation]]s of the plane. A [[sequence (mathematics)|sequence]] can also be viewed as a function defined on the [[natural number]]s, and for a [[periodic sequence]] these notions are defined accordingly.

==Examples==
[[Image:Sine.svg|thumb|right|350px|A graph of the sine function, showing two complete periods]]

===Real number examples===
For example, the [[sine function]] is periodic with period <math>2\pi</math>, since

:<math>\sin(x + 2\pi) = \sin x</math>

for all values of <math>x</math>.  This function repeats on intervals of length <math>2\pi</math> (see the graph to the right).

Everyday examples are seen when the variable is ''time''; for instance the hands of a [[clock]] or the phases of the [[moon]] show periodic behaviour. '''Periodic motion''' is motion in which the position(s) of the system are expressible as periodic functions, all with the ''same'' period.

For a function on the [[real number]]s or on the [[integer]]s, that means that the entire [[Graph of a function|graph]] can be formed from copies of one particular portion, repeated at regular intervals.

A simple example of a periodic function is the function <math>f</math> that gives the "[[fractional part]]" of its argument. Its period is 1. In particular,

: <math>f(0.5) = f(1.5) = f(2.5) = \cdots = 0.5</math> 

The graph of the function <math>f</math> is the [[sawtooth wave]].

[[Image:Sine cosine plot.svg|300px|right|thumb|A plot of <math>f(x) = \sin(x)</math> and <math>g(x) = \cos(x)</math>; both functions are periodic with period 2π.]]
The [[trigonometric function]]s sine and cosine are common periodic functions, with period 2π (see the figure on the right).  The subject of [[Fourier series]] investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.

According to the definition above, some exotic functions, for example the [[Dirichlet function]], are also periodic; in the case of Dirichlet function, any nonzero rational number is a period.

===Complex number examples===
Using [[complex analysis|complex variables]] we have the common period function:

:<math>e^{ikx} = \cos kx + i\,\sin kx.</math>

Since the cosine and sine functions are both periodic with period 2π, and the complex exponential above is made up of cosine and sine waves, the above (actually [[Euler's formula]]) has the following property.  If ''L'' is the period of the function then

:<math>L = \frac{2\pi}{k}.</math>

Complex functions may be periodic along one line or axis in the complex plane but not on another. For instance, <math>e^{z}</math> is periodic along the imaginary axis but not the real axis.

====Double-periodic functions====
A function whose domain is the [[complex number]]s can have two incommensurate periods without being constant. The [[elliptic function]]s are such functions. ("Incommensurate" in this context means not real multiples of each other.)

==Properties==
<!-- '''periodicity with period zero''' ''P'' ''' greater than zero if !-->
Periodic functions can take on values many times. More specifically, if a function <math>f</math> is periodic with period <math>P</math>, then for all <math>x</math> in the domain of <math>f</math> and all positive integers <math>n</math>,

: <math>f(x + nP) = f(x)</math>

If <math>f(x)</math> is a function with period <math>P</math>, then <math>f(ax)</math>, where <math>a</math> is a non-zero real number such that <math>ax</math> is within the domain of <math>f</math>, is periodic with period <math display="inline">\frac{P}{|a|}</math>. For example, <math>f(x) = \sin(x)</math> has period <math>2 \pi</math> therefore <math>\sin(5x)</math> will have period <math display="inline">\frac{2\pi}{5}</math>.

Some periodic functions can be described by [[Fourier series]]. For instance, for [[Lp space|''L''<sup>2</sup> functions]], [[Carleson's theorem]] states that they have a [[pointwise]] ([[Lebesgue measure|Lebesgue]]) [[almost everywhere convergence|almost everywhere convergent]] [[Fourier series]]. Fourier series can only be used for periodic functions, or for functions on a bounded (compact) interval. If <math>f</math> is a periodic function with period <math>P</math> that can be described by a Fourier series, the coefficients of the series can be described by an integral over an interval of length <math>P</math>.

==Generalizations==

===Antiperiodic functions===
One common subset of periodic functions is that of '''antiperiodic functions'''.  This is a function ''f'' such that ''f''(''x''&nbsp;+&nbsp;''P'') = &minus;''f''(''x'') for all ''x''.  (Thus, a ''P''-antiperiodic function is a 2''P''-periodic function.) For example, the sine or cosine function is π-antiperiodic and 2π-periodic. While a ''P''-antiperiodic function is a 2''P''-periodic function, the inverse is not necessarily true.

===Bloch-periodic functions===
A further generalization appears in the context of [[Bloch wave]]s and [[Floquet theory]], which govern the solution of various periodic differential equations.  In this context, the solution (in one dimension) is typically a function of the form:

:<math>f(x+P) = e^{ikP} f(x) </math>

where ''k'' is a real or complex number (the ''Bloch wavevector'' or ''Floquet exponent'').  Functions of this form are sometimes called '''Bloch-periodic''' in this context.   A periodic function is the special case ''k''&nbsp;=&nbsp;0, and an antiperiodic function is the special case ''k''&nbsp;=&nbsp;π/''P''.

===Quotient spaces as domain===
In [[signal processing]] you encounter the problem, that [[Fourier series]] represent periodic functions and that Fourier series satisfy [[convolution theorem]]s (i.e. [[convolution]] of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this end you can use the notion of a [[quotient space (linear algebra)|quotient space]]:

:<math>{\mathbb{R}/\mathbb{Z}}
 = \{x+\mathbb{Z} : x\in\mathbb{R}\}
 = \{\{y : y\in\mathbb{R}\land y-x\in\mathbb{Z}\} : x\in\mathbb{R}\}</math>.

That is, each element in <math>{\mathbb{R}/\mathbb{Z}}</math> is an [[equivalence class]] of [[real number]]s that share the same [[fractional part]]. Thus a function like <math>f : {\mathbb{R}/\mathbb{Z}}\to\mathbb{R}</math> is a representation of a 1-periodic function.

==Calculating period==
Consider a real waveform consisting of superimposed frequencies, expressed in a set as ratios to a fundamental frequency, f: F = {{frac|1|f}}{{nnbsp}}[f{{sub|1}} f{{sub|2}} f{{sub|3}} … f{{sub|N}}] where all non-zero elements ≥1 and at least one of the elements of the set is 1. To find the period, T, first find the least common denominator of all the elements in the set. Period can be found as T = {{frac|LCD|f}}. Consider that for a simple sinusoid, T = {{frac|1|f}}. Therefore, the LCD can be seen as a periodicity multiplier. 

* For set representing all notes of Western major scale: [1 {{frac|9|8}} {{frac|5|4}} {{frac|4|3}} {{frac|3|2}} {{frac|5|3}} {{frac|15|8}}] the LCD is 24 therefore T = {{frac|24|f}}.
* For set representing all notes of a major triad: [1 {{frac|5|4}} {{frac|3|2}}] the LCD is 4 therefore T = {{frac|4|f}}.
* For set representing all notes of a minor triad: [1 {{frac|6|5}} {{frac|3|2}}] the LCD is 10 therefore T = {{frac|10|f}}.

If no least common denominator exists, such that if one of the above elements were irrational, then the wave would not be periodic.<ref>https://www.ece.rice.edu/~srs1/files/Lec6.pdf</ref>

==See also==
{{Div col|colwidth=25em}}
* [[List of periodic functions]]
* [[Periodic sequence]]
* [[Almost periodic function]]
* [[Amplitude]]
* [[Definite pitch]]
* [[Doubly periodic function]]
* [[Frequency]]
* [[Oscillation]]
* [[Quasiperiodic function]]
* [[Wavelength]]
* [[Periodic summation]]
* [[Secular variation]]
{{div col end}}

==References==
{{Reflist}}
* {{cite book|last=Ekeland|first=Ivar|authorlink=Ivar Ekeland|chapter=One|title=Convexity methods in Hamiltonian mechanics|series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]|volume=19|publisher=Springer-Verlag|location=Berlin|year=1990|pages=x+247|isbn=3-540-50613-6|mr=1051888|ref=harv}}

==External links==
* {{springer|title=Periodic function|id=p/p072170}}
*[http://mathworld.wolfram.com/PeriodicFunction.html Periodic functions at MathWorld]

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[[Category:Calculus]]
[[Category:Elementary mathematics]]
[[Category:Fourier analysis]]
[[Category:Types of functions]]