Editing Periodic function (section)

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

===Real number examples===
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 <math>2\pi</math>.]]
The [[trigonometric function]]s sine and cosine are common periodic functions, with period <math>2\pi</math> (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 <math>2\pi</math>, the complex exponential is made up of cosine and sine waves. This means that [[Euler's formula]] (above) has the property such that if <math>L</math> is the period of the function, then

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

====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.)