Periodic function

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In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. 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 numbers or on the integers, that means that the entire 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 + Pn ) = f ( x ).

In the above example, the value of P is 1, since f( x ) = f( x + 1 ) = f( x + 2 ) = etc. The period of a function need not be the smallest value (least period) that satisfies the above equation, so P could also equal two.

A plot of the f(x) = sin(x) and f(x) = cos(x) functions, both with period 2Pi.

A simple example is the function f that gives the "fractional part" of its argument:

f( 0.5 ) = f( 1.5 ) = f( 2.5 ) = ... = 0.5.

Some named examples are sawtooth wave, square wave and triangle wave.

The trigonometric functions sine and cosine are common periodic functions, with period 2π. 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 numbers can have two incommensurate periods without being constant. The elliptic functions are such functions. ("Incommensurate" in this context means not real multiples of each other.)

1 General definition
2 Antiperiodic functions and other generalizations
3 Periodic sequences
4 Translational symmetry
5 See also
6 External links

[edit] General definition

Let E be a set with an internal operation + . A T-periodic function, or function periodic with period T on E is a function f on E to some set F, such that

for all x in E, f(x + T) = f(x).

Note that unless + is assumed commutative this definition depends on writing T on the right.

The period T is not unique. For a given T, every integer multiple of T is also a period.

[edit] 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 + T) = −f(x) for all x. (Thus, a T-antiperiodic function is a 2T-periodic function.)

A further generalization appears in the context of Bloch waves 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:

f (x + T) = e i k T f (x)

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/T.

[edit] Periodic sequences

Some naturally-occurring sequences 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.

[edit] 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.