Periodic function
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2008-07-11T16:45:38Z
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In [[mathematics]], a '''periodic function''' is a [[function (mathematics)|function]] that repeats its values after some definite ''period'' has been added to its [[independent variable]]. This property is called [[periodicity]].
[[Image:Periodic function illustration.svg|thumb|right|300px|An illustration of a periodic function with period <math>P.</math>]]
==Examples==
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. More explicitly, a function ''f'' is '''periodic with period ''P'' ''' greater than zero if
: ''f''(''x'' + ''P'') = ''f''(''x'')
for ''all'' values of ''x'' in the domain of ''f''. An '''aperiodic function''' (non-periodic function) is one that has no such period ''P'' (not to be confused with an '''antiperiodic function''' (below) for which ''f''(''x'' + ''P'') = −''f''(''x'') for some ''P'').
If a function ''f'' is periodic with period ''P'', then for all ''x'' in the domain of ''f'' and all integers ''n'',
: ''f''(''x'' + ''nP'') = ''f''(''x'').
[[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.]]
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,
: ''f''( 0.5 ) = ''f''( 1.5 ) = ''f''( 2.5 ) = ... = 0.5.
The graph of the function ''f'' is the [[sawtooth wave]].
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.
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==
if ''f''(''x'') is a function with period ''P'', then ''f''(''ax''), 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.
== Antiperiodic functions and other generalizations ==
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.)
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'' = 0, and an antiperiodic function is the special case ''k'' = π/''P''.
==Periodic sequences==<!-- This section is linked from [[Amicable number]] -->
Some naturally-occurring [[sequence]]s are periodic, for example (eventually) the [[decimal]] expansion of any [[rational number]] (see [[recurring decimal]]). We can therefore speak of the '''period''' or '''period length''' of a sequence. This is (if one insists) just a special case of the general definition.
==Periodic mapping==
A mapping ''f'' of a set into itself is said to be periodic if some iterate ''f''<sup>''n''</sup> is the identity mapping for some integer ''n'' > 1; the smallest possible ''n'' is called the ''period'' of ''f''. This concept is commonly used in the theory of [[dynamical system]]s.
==Translational symmetry==
If a function is used to describe an object, e.g. an infinite image is given by the color as function of position, the periodicity of the function corresponds to [[translational symmetry]] of the object.
==Cycle==
The restriction of a periodic function to an interval whose length is equal to the period is called a '''cycle'''.
==See also==
* [[Almost periodic function]]
* [[Amplitude]]
* [[Definite pitch]]
* [[Doubly-periodic function]]
* [[Frequency]]
* [[Oscillation]]
* [[Quasiperiodic function]]
* [[Wavelength]]
==External links==
*[http://mathworld.wolfram.com/PeriodicFunction.html Periodic functions at MathWorld]
[[Category:Calculus]]
[[Category:Elementary mathematics]]
[[Category:Fourier analysis]]
[[Category:Functions and mappings]]
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