A periodic function is a
function 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 all ...
, which repeat at intervals of
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 c ...
s, are periodic functions. Periodic functions are used throughout science to describe
oscillation
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
s,
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 (res ...
s, and other phenomena that exhibit
periodicity
Periodicity or periodic may refer to:
Mathematics
* Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups
* Periodic function, a function whose output contains values tha ...
. 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
:
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 equatio ...
, 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 iteratio ...
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 (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...
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 n ...
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 opp ...
is periodic with period
, since
:
for all values of
. This function repeats on intervals of length
(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 real ...
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 ...
s, that means that the entire
graph can be formed from copies of one particular portion, repeated at regular intervals.
A simple example of a periodic function is the function
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,
:
The graph of the function
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 function
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 all ...
s sine and cosine are common periodic functions, with period
(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, are also periodic; in the case of Dirichlet function, any nonzero rational number is a period.
Complex number examples
Using
complex variables
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
we have the common period function:
:
Since the cosine and sine functions are both periodic with period
, 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 an ...
(above) has the property such that if
is the period of the function, then
:
Double-periodic functions
A function whose domain is the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s 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 in ...
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
is periodic with period
, then for all
in the domain of
and all positive integers
,
:
If
is a function with period
, then
, where
is a non-zero real number such that
is within the domain of
, is periodic with period
. For example,
has period
therefore
will have period
.
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 res ...
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
is a periodic function with period
that can be described by a Fourier series, the coefficients of the series can be described by an integral over an interval of length
.
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
).
Generalizations
Antiperiodic functions
One subset of periodic functions is that of antiperiodic functions. This is a function
such that
for all
. For example, the sine and cosine functions are
-antiperiodic and
-periodic. While a
-antiperiodic function is a
-periodic function, the
converse is not necessarily true.
Bloch-periodic functions
A further generalization appears in the context of
Bloch's theorems and
Floquet theory
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form
:\dot = A(t) x,
with \displaystyle A(t) a Piecewise#Continuity, piecewise contin ...
, which govern the solution of various periodic differential equations. In this context, the solution (in one dimension) is typically a function of the form
:
where
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
, and an antiperiodic function is the special case
. Whenever
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 audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniq ...
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 theorems (i.e.
convolution
In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions ( and ) that produces a third function (f*g) that expresses how the shape of one is ...
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 a ...
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 real ...
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
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