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A periodic function is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
that repeats its values at regular intervals. For example, the
trigonometric functions In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
, which repeat at intervals of 2\pi
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s, are periodic functions. Periodic functions are used throughout science to describe oscillations,
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
s, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.


Definition

A function is said to be periodic if, for some nonzero constant , it is the case that :f(x+P) = f(x) for all values of in the domain. A nonzero constant for which this is the case is called a period of the function. If there exists a least positive constant with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function is used to mean its fundamental period. A function with period will repeat on intervals of length , and these intervals are sometimes also referred to as periods of the function. Geometrically, a periodic function can be defined as a function whose graph exhibits
translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
, i.e. a function is periodic with period if the graph of is
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
under
translation Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
in the -direction by a distance of . This definition of periodicity can be extended to other geometric shapes and patterns, as well as be generalized to higher dimensions, such as periodic
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
s of the plane. A
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
can also be viewed as a function defined on the
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...
s, and for a
periodic sequence In mathematics, a periodic sequence (sometimes called a cycle) is a sequence for which the same terms are repeated over and over: :''a''1, ''a''2, ..., ''a'p'',  ''a''1, ''a''2, ..., ''a'p'',  ''a''1, ''a''2, ..., ''a' ...
these notions are defined accordingly.


Examples


Real number examples

The
sine function In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is op ...
is periodic with period 2\pi, since :\sin(x + 2\pi) = \sin x for all values of x. This function repeats on intervals of length 2\pi (see the graph to the right). Everyday examples are seen when the variable is ''time''; for instance the hands of a
clock A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and the ...
or the phases of the
moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...
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 In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
s or on the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
s, that means that the entire
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties * Graph (topology), a topological space resembling a graph in the sense of discr ...
can be formed from copies of one particular portion, repeated at regular intervals. A simple example of a periodic function is the function f that gives the "
fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
" of its argument. Its period is 1. In particular, : f(0.5) = f(1.5) = f(2.5) = \cdots = 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. It is so named based on its resemblance to the teeth of a plain-toothed saw with a zero rake angle. A single sawtooth, or an intermittently triggered sawtooth, is called a ...
. The trigonometric functions sine and cosine are common periodic functions, with period 2\pi (see the figure on the right). The subject of
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
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 In mathematics, the Dirichlet function is the indicator function 1Q or \mathbf_\Q of the set of rational numbers Q, i.e. if ''x'' is a rational number and if ''x'' is not a rational number (i.e. an irrational number). \mathbf 1_\Q(x) = \begin 1 ...
, are also periodic; in the case of Dirichlet function, any nonzero rational number is a period.


Complex number examples

Using complex variables we have the common period function: :e^ = \cos kx + i\,\sin kx. Since the cosine and sine functions are both periodic with period 2\pi, the complex exponential is made up of cosine and sine waves. This means that
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for a ...
(above) has the property such that if L is the period of the function, then :L = \frac.


Double-periodic functions

A function whose domain is the complex numbers can have two incommensurate periods without being constant. The
elliptic function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
s are such functions. ("Incommensurate" in this context means not real multiples of each other.)


Properties

Periodic functions can take on values many times. More specifically, if a function f is periodic with period P, then for all x in the domain of f and all positive integers n, : f(x + nP) = f(x) If f(x) is a function with period P, then f(ax), where a is a non-zero real number such that ax is within the domain of f, is periodic with period \frac. For example, f(x) = \sin(x) has period 2 \pi therefore \sin(5x) will have period \frac. Some periodic functions can be described by
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
. For instance, for ''L''2 functions,
Carleson's theorem Carleson's theorem is a fundamental result in mathematical analysis establishing the pointwise (Lebesgue) almost everywhere convergence of Fourier series of functions, proved by . The name is also often used to refer to the extension of the re ...
states that they have a
pointwise In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
(
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
) almost everywhere convergent
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
. Fourier series can only be used for periodic functions, or for functions on a bounded (compact) interval. If f is a periodic function with period P that can be described by a Fourier series, the coefficients of the series can be described by an integral over an interval of length P. Any function that consists only of periodic functions with the same period is also periodic (with period equal or smaller), including: * addition, subtraction, multiplication and division of periodic functions, and * taking a power or a root of a periodic function (provided it is defined for all x).


Generalizations


Antiperiodic functions

One subset of periodic functions is that of antiperiodic functions. This is a function f such that f(x+P) = -f(x) for all x. For example, the sine and cosine functions are \pi-antiperiodic and 2\pi-periodic. While a P-antiperiodic function is a 2P-periodic function, the
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical ...
is not necessarily true.


Bloch-periodic functions

A further generalization appears in the context of
Bloch's theorem In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who di ...
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 :f(x+P) = e^ 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=\pi/P. Whenever k P/ \pi is rational, the function is also periodic.


Quotient spaces as domain

In
signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, d ...
you encounter the problem, that
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
represent periodic functions and that Fourier series satisfy
convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g ...
s (i.e.
convolution In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' ...
of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this end you can use the notion of a quotient space: : = \ = \. That is, each element in is an
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every r ...
s that share the same
fractional part The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
. Thus a function like f : \to\mathbb is a representation of a 1-periodic function.


Calculating period

Consider a real waveform consisting of superimposed frequencies, expressed in a set as ratios to a fundamental frequency, f: F = f f ... fwhere all non-zero elements ≥1 and at least one of the elements of the set is 1. To find the period, T, first find the least common denominator of all the elements in the set. Period can be found as T = . Consider that for a simple sinusoid, T = . Therefore, the LCD can be seen as a periodicity multiplier. * For set representing all notes of Western major scale: the LCD is 24 therefore T = . * For set representing all notes of a major triad: the LCD is 4 therefore T = . * For set representing all notes of a minor triad: the LCD is 10 therefore T = . If no least common denominator exists, for instance if one of the above elements were irrational, then the wave would not be periodic.


See also


References

*


External links

*
Periodic functions at MathWorld
{{Authority control Calculus Elementary mathematics Fourier analysis Types of functions