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Periodic function

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From Wikipedia, the free encyclopedia

Function that repeats its values at regular intervals or periods

Not to be confused withperiodic mapping.

"Period length" redirects here; not to be confused withrepeating decimal.

"Aperiodic" and "Non-periodic" redirect here. For other uses, seeAperiodic (disambiguation).

An illustration of a periodic function with period $P . {\displaystyle P.}$

An illustration of a periodic function with period $P . {\displaystyle P.}$

Aperiodic function, also called aperiodic waveform (or simplyperiodic wave), is afunction that repeats its values at regular intervals orperiods. The repeatable part of the function orwaveform is called acycle.^[1] For example, thetrigonometric functions, which repeat at intervals of $2\pi$ radians, are periodic functions. Periodic functions are used throughout science to describeoscillations,waves, and other phenomena that exhibitperiodicity. Any function that is not periodic is calledaperiodic.

Definition

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A functionf is said to beperiodic if, for somenonzero constantP, it is the case that

f(x+P)=f(x)

for all values ofx in the domain. A nonzero constantP for which this is the case is called aperiod of the function. If there exists a least positive^[2] constantP with this property, it is called thefundamental period (alsoprimitive period,basic period, orprime period.) Often, "the" period of a function is used to mean its fundamental period. A function with periodP will repeat on intervals of lengthP, and these intervals are sometimes also referred to asperiods of the function.

Geometrically, a periodic function can be defined as a function whose graph exhibitstranslational symmetry, i.e. a functionf is periodic with periodP if the graph off isinvariant undertranslation in thex-direction by a distance ofP. This definition of periodicity can be extended to other geometric shapes and patterns, as well as be generalized to higher dimensions, such as periodictessellations of the plane. Asequence can also be viewed as a function defined on thenatural numbers, and for aperiodic sequence these notions are defined accordingly.

Examples

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A graph of the sine function, showing two complete periods

Real number examples

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Thesine function is periodic with period $2\pi$ , since

\sin(x+2\pi )=\sin x

for all values of $x {\displaystyle x}$ . This function repeats on intervals of length $2\pi$ (see the graph to the right).

Everyday examples are seen when the variable istime; for instance the hands of aclock or the phases of themoon show periodic behaviour.Periodic motion is motion in which the position(s) of the system are expressible as periodic functions, all with thesame period.

For a function on thereal numbers or on theintegers, that means that the entiregraph can be formed from copies of one particular portion, repeated at regular intervals.

A simple example of a periodic function is the function $f {\displaystyle f}$ that gives the "fractional part" 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 $f {\displaystyle f}$ is thesawtooth wave.

A plot of $f(x)=\sin(x)$ and $g(x)=\cos(x)$ ; both functions are periodic with period $2\pi$ .

{\displaystyle f(x)=\sin(x)} — A plot of $f(x)=\sin(x)$ and $g(x)=\cos(x)$ ; both functions are periodic with period $2\pi$ .

Thetrigonometric functions sine and cosine are common periodic functions, with period $2\pi$ (see the figure on the right). The subject ofFourier series 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 theDirichlet function, are also periodic; in the case of Dirichlet function, any nonzero rational number is a period.

Complex number examples

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Usingcomplex variables we have the common period function:

e^{ikx}=\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 thatEuler's formula (above) has the property such that if $L {\displaystyle L}$ is the period of the function, then

L={\frac {2\pi }{k}}.

Double-periodic functions

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A function whose domain is thecomplex numbers can have two incommensurate periods without being constant. Theelliptic functions are such functions. ("Incommensurate" in this context means not real multiples of each other.)

Properties

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Periodic functions can take on values many times. More specifically, if a function $f {\displaystyle f}$ is periodic with period $P {\displaystyle P}$ , then for all $x {\displaystyle x}$ in the domain of $f {\displaystyle f}$ and all positive integers $n {\displaystyle n}$ ,

f(x+nP)=f(x)

If $f(x)$ is a function with period $P {\displaystyle P}$ , then $f(ax)$ , where $a {\displaystyle a}$ is a non-zero real number such that $a x {\displaystyle ax}$ is within the domain of $f {\displaystyle f}$ , is periodic with period ${\textstyle {\frac {P}{a}}}$ . For example, $f(x)=\sin(x)$ has period $2\pi$ and, therefore, $\sin(5x)$ will have period ${\textstyle {\frac {2\pi }{5}}}$ .

Some periodic functions can be described byFourier series. For instance, forL² functions,Carleson's theorem states that they have apointwise (Lebesgue)almost everywhere convergent Fourier series. Fourier series can only be used for periodic functions, or for functions on a bounded (compact) interval. If $f {\displaystyle f}$ is a periodic function with period $P {\displaystyle P}$ that can be described by a Fourier series, the coefficients of the series can be described by an integral over an interval of length $P {\displaystyle P}$ .

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 $x {\displaystyle x}$ ).

Generalizations

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Antiperiodic functions

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One subset of periodic functions is that ofantiperiodic functions. This is a function $f {\displaystyle f}$ such that $f(x+P)=-f(x)$ for all $x {\displaystyle x}$ . For example, the sine and cosine functions are $\pi$ -antiperiodic and $2\pi$ -periodic. While a $P {\displaystyle P}$ -antiperiodic function is a $2P$ -periodic function, theconverse is not necessarily true.^[3]

Bloch-periodic functions

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A further generalization appears in the context ofBloch's theorems andFloquet 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^{ikP}f(x)~,

where $k {\displaystyle k}$ is a real or complex number (theBloch wavevector orFloquet exponent). Functions of this form are sometimes calledBloch-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 $kP/\pi$ is rational, the function is also periodic.

Quotient spaces as domain

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Insignal processing you encounter the problem, thatFourier series represent periodic functions and that Fourier series satisfyconvolution theorems (i.e.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 aquotient space:

{\mathbb {R} /\mathbb {Z} }=\{x+\mathbb {Z} :x\in \mathbb {R} \}=\{\{y:y\in \mathbb {R} \land y-x\in \mathbb {Z} \}:x\in \mathbb {R} \}

.

That is, each element in ${\mathbb {R} /\mathbb {Z} }$ is anequivalence class ofreal numbers that share the samefractional part. Thus a function like $f:{\mathbb {R} /\mathbb {Z} }\to \mathbb {R}$ is a representation of a 1-periodic function.

Calculating period

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Consider a real waveform consisting of superimposed frequencies, expressed in a set as ratios to afundamental frequency, f: F =1⁄f [f₁ f₂ f₃ ... f_N] where 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 =LCD⁄f. Consider that for a simple sinusoid, T =1⁄f. Therefore, the LCD can be seen as a periodicity multiplier.

For set representing all notes of Westernmajor scale: [19⁄85⁄44⁄33⁄25⁄315⁄8] the LCD is 24 therefore T =24⁄f.
For set representing all notes of a major triad: [15⁄43⁄2] the LCD is 4 therefore T =4⁄f.
For set representing all notes of a minor triad: [16⁄53⁄2] the LCD is 10 therefore T =10⁄f.

If no least common denominator exists, for instance if one of the above elements were irrational, then the wave would not be periodic.^[4]

References

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^"IEC 60050 — Details for IEV number 103-05-08: "cycle"".International Electrotechnical Vocabulary. Retrieved2023-11-20.
^For some functions, like aconstant function or theDirichlet function (theindicator function of therational numbers), a least positive period may not exist (theinfimum of all positive periodsP being zero).
^Weisstein, Eric W."Antiperiodic Function".mathworld.wolfram.com. Retrieved2024-06-06.
^Summerson, Samantha R. (5 October 2009)."Periodicity, Real Fourier Series, and Fourier Transforms"(PDF). Archived fromthe original(PDF) on 2019-08-25. Retrieved2018-03-24.

Ekeland, Ivar (1990). "One".Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Vol. 19. Berlin: Springer-Verlag. pp. x+247.ISBN 3-540-50613-6.MR 1051888.