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| {{Distinguish2|periodic mapping, a mapping whose nth iterate is the identity (see [[periodic point]])}}
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| 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 2''π'' [[radian]]s. Periodic functions are used throughout science to describe [[oscillation]]s, [[wave]]s, and other phenomena that exhibit [[Frequency|periodicity]]. Any function which is not periodic is called '''aperiodic'''.
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| [[Image:Periodic function illustration.svg|thumb|right|300px|An illustration of a periodic function with period <math>P.</math>]]
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| == Definition ==
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| A function ''f'' is said to be '''periodic''' with period ''P'' (''P'' being a nonzero constant) if we have
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| :<math>f(x+P) = f(x) \,\!</math>
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| for all values of ''x''. 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 possible positive ''P'' being zero).</ref>
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| constant ''P'' with this property, it is called the '''fundamental period''' (also '''primitive period''', '''basic period''', or '''prime period'''.) A function with period ''P'' will repeat on intervals of length ''P'', and these intervals
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| are referred to as '''periods'''.
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| Geometrically, a periodic function can be defined as a function whose graph exhibits [[translational symmetry]]. Specifically, a function ''f'' is periodic with period ''P'' if the graph of ''f'' is [[invariant (mathematics)|invariant]] under [[translation (geometry)|translation]] in the ''x''-direction by a distance of ''P''. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic [[tessellation]]s of the plane.
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| A function that is not periodic is called '''aperiodic'''.
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| ==Examples==
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| [[Image:Sine.svg|thumb|right|350px|A graph of the sine function, showing two complete periods]]
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| For example, the [[sine function]] is periodic with period 2''π'', since
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| :<math>\sin(x + 2\pi) = \sin x \,\!</math>
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| for all values of ''x''. This function repeats on intervals of length 2''π'' (see the graph to the right).
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| 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.
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| 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.
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| A simple example of a periodic function is the function ''f'' that gives the "[[fractional part]]" of its argument. Its period is 1. In particular,
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| : ''f''( 0.5 ) = ''f''( 1.5 ) = ''f''( 2.5 ) = ... = 0.5.
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| The graph of the function ''f'' is the [[sawtooth wave]].
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| [[Image:Sine cosine plot.svg|300px|right|thumb|A plot of ''f''(''x'') = sin(''x'') and ''g''(''x'') = cos(''x''); both functions are periodic with period 2π.]]
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| 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.
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| 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.
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| ==Properties==
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| <!-- '''periodicity with period zero''' ''P'' ''' greater than zero if !-->
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| If a function ''f'' is periodic with period ''P'', then for all ''x'' in the domain of ''f'' and all integers ''n'',
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| : ''f''(''x'' + ''nP'') = ''f''(''x''). | |
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| If ''f''(''x'') is a function with period ''P'', then ''f''(''ax+b''), where ''a'' is a positive constant, is periodic with period ''P/|a|''. For example, ''f''(''x'')=sin''x'' has period 2π, therefore sin(5''x'') will have period 2π/5.
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| ==Double-periodic functions==
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| A function whose domain is the [[complex number]]s can have two incommensurate periods without being constant. The [[elliptic function]]s are such functions.
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| ("Incommensurate" in this context means not real multiples of each other.)
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| ==Complex example==
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| Using [[complex analysis|complex variables]] we have the common period function:
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| :<math>e^{ikx} = \cos kx + i\,\sin kx</math>
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| As you can see, since the cosine and sine functions are periodic, and the complex exponential above is made up of cosine/sine waves, then the above (actually [[Euler's formula]]) has the following property. If ''L'' is the period of the function then:
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| :<math>L = 2\pi/k </math>
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| == Generalizations ==
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| === Antiperiodic functions ===
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| One common generalization of periodic functions is that of '''antiperiodic functions'''. This is a function ''f'' such that ''f''(''x'' + ''P'') = −''f''(''x'') for all ''x''. (Thus, a ''P''-antiperiodic function is a 2''P''-periodic function.)
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| === Bloch-periodic functions ===
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| 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:
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| :<math>f(x+P) = e^{ikP} f(x) \,\!</math> | |
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| 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'' = 0, and an antiperiodic function is the special case ''k'' = π/''P''.
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| === Quotient spaces as domain ===
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| In [[signal processing]] you encounter the problem, that [[Fourier series]] represent periodic functions
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| and that Fourier series satisfy [[convolution theorem]]s
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| (i.e. [[convolution]] of Fourier series corresponds to multiplication of represented periodic function and vice versa),
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| but periodic functions cannot be convolved with the usual definition,
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| since the involved integrals diverge.
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| A possible way out is to define a periodic function on a bounded but periodic domain.
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| To this end you can use the notion of a [[Quotient space (linear algebra)|quotient space]]:
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| :<math>{\mathbb{R}/\mathbb{Z}}
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| = \{x+\mathbb{Z} : x\in\mathbb{R}\}
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| = \{\{y : y\in\mathbb{R}\land y-x\in\mathbb{Z}\} : x\in\mathbb{R}\}</math>.
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| That is, each element in <math>{\mathbb{R}/\mathbb{Z}}</math> is an [[equivalence class]]
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| of [[real number]]s that share the same [[fractional part]].
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| Thus a function like <math>f : {\mathbb{R}/\mathbb{Z}}\to\mathbb{R}</math>
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| is a representation of a 1-periodic function.
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| ==See also==
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| * [[List of periodic functions]]
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| * [[Periodic sequence]]
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| * [[Almost periodic function]]
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| * [[Amplitude]]
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| * [[Definite pitch]]
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| * [[Doubly periodic function]]
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| * [[Floquet theory]]
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| * [[Frequency]]
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| * [[Oscillation]]
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| * [[Quasiperiodic function]]
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| * [[Wavelength]]
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| * [[Periodic summation]]
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| * [[Secular variation]]
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| ==References==
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| {{Reflist}}
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| * {{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}}
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| ==External links==
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| * {{springer|title=Periodic function|id=p/p072170}}
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| *[http://mathworld.wolfram.com/PeriodicFunction.html Periodic functions at MathWorld]
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| [[Category:Calculus]]
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| [[Category:Elementary mathematics]]
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| [[Category:Fourier analysis]]
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| [[Category:Types of functions]]
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