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, which repeat at intervals of
radians, are periodic functions. Periodic functions are used throughout science to describe
oscillations,
waves, 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, i.e. a function is periodic with period if the graph of is
invariant under
translation 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
tessellations of the plane. A
sequence can also be viewed as a function defined on the
natural numbers, and for a
periodic sequence these notions are defined accordingly.
Examples
Real number examples
The
sine function 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 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.
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
trigonometric functions 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 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 a ...
(above) has the property such that if
is the period of the function, then
:
Double-periodic functions
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.)
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 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)
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 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
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 continuous periodic function ...
, 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 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 of
real numbers 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
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Calculus
Elementary mathematics
Fourier analysis
Types of functions