In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Dependent variables and independent variables refer to values that change in relationship to each other This property is called periodicity. Periodicity is the quality of occurring at regular intervals or periods (in Time or Space) and can occur in different contexts A Clock marks

An illustration of a periodic function with period P.

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. Circadian Locomotor Output Cycles Kaput, or Clock is a gene which encodes proteins regulating Circadian rhythm. 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. In Mathematics, the real numbers may be described informally in several different ways The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In mathematics the graph of a function f is the collection of all Ordered pairs ( x, f ( x) 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).

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 sawtooth wave (or saw wave) is a kind of Non-sinusoidal waveform.

The trigonometric functions 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. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions

A function whose domain is the complex numbers can have two incommensurate periods without being constant. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted The elliptic functions are such functions. In Complex analysis, an elliptic function is a function defined on the Complex plane which is periodic in two directions (a Doubly-periodic ("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)=sinx has period 2π, therefore sin(5x) 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 2P-periodic function. )

A further generalization appears in the context of Bloch waves and Floquet theory, which govern the solution of various periodic differential equations. A Bloch wave or Bloch state, named after Felix Bloch, is the Wavefunction of a particle (usually an Electron) placed in a periodic potential Floquet theory is a branch of the theory of Ordinary differential equations relating to the class of solutions to Linear differential equations of the form In this context, the solution (in one dimension) is typically a function of the form:

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

Some naturally-occurring sequences are periodic, for example (eventually) the decimal expansion of any rational number (see recurring decimal). In Mathematics, a sequence is an ordered list of objects (or events The decimal ( base ten or occasionally denary) Numeral system has ten as its base. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions A Decimal representation of a Real number is called a repeating decimal (or recurring decimal) if at some point it becomes periodic: there is 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 fⁿ 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 systems. The dynamical system concept is a mathematical Formalization for any fixed "rule" which describes the Time dependence of a point's position

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. In Geometry, a translation "slides" an object by a vector a: T a (p = p + a

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
• Frequency
• Oscillation
• Quasiperiodic function
• Wavelength

External links

• Periodic functions at MathWorld