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Periodic function

From Wikipedia, the free encyclopedia

Function with a repeating pattern

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 $P . {\displaystyle P.}$

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

Aperiodic function is afunction that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describewaves and other repeating phenomena, are periodic. Many aspects of the natural world have periodic behavior, such as thephases of the Moon, the swinging of apendulum, and thebeating of a heart.

The length of the interval over which a periodic function repeats is called itsperiod. Any function that is not periodic is calledaperiodic.

Definition

A graph of the sine function. It is periodic with a fundamental period of $2\pi$ .

{\displaystyle 2\pi } — A graph of the sine function. It is periodic with a fundamental period of $2\pi$ .

A function is defined asperiodic if its values repeat at regular intervals. For example, the positions of the hands on aclock display periodic behavior as they cycle through the same positions every 12 hours. This repeating interval is known as theperiod.

More formally, a function $f {\displaystyle f}$ is periodic if there exists a constant $P {\displaystyle P}$ such that

f(x+P)=f(x)

for all values of $x {\displaystyle x}$ in thedomain. Anonzero constant $P {\displaystyle P}$ for which this condition holds is called aperiod of the function.^[1]

If a period $P {\displaystyle P}$ exists, any integer multiple $n P {\displaystyle nP}$ (for a positive integer $n {\displaystyle n}$ ) is also a period. If there is aleast positive period, it is called thefundamental period (alsoprimitive period orbasic period).^[2] Often, "the" period of a function is used to refer to its fundamental period.

Geometrically, a periodic function's graph exhibitstranslational symmetry. Its graph isinvariant undertranslation in the $x {\displaystyle x}$ -direction by a distance of $P {\displaystyle P}$ . This implies that the entire graph can be formed from copies of one particular portion, repeated at regular intervals.

Examples

Periodic behavior can be illustrated through both common, everyday examples and more formal mathematical functions.

Real-valued functions

Functions that map real numbers to real numbers can display periodicity, which is often visualized on a graph.

Sawtooth wave

An example is the function $f {\displaystyle f}$ that represents the "fractional part" of its argument. Its period is 1. For instance,

f(0.5)=f(1.5)=f(2.5)=\cdots =0.5

The graph of the function $f {\displaystyle f}$ is asawtooth wave.

Trigonometric functions

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 $2\pi$ .

The trigonometric functions are common examples of periodic functions. Thesine function andcosine function are periodic with a fundamental period of $2\pi$ , as illustrated in the figure to the right. For the sine function, this is expressed as:

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

for all values of $x {\displaystyle x}$ .

The field ofFourier series investigates the concept that an arbitrary periodic function can be expressed as a sum of trigonometric functions with matching periods.

Exotic functions

Some functions are periodic but possess properties that make them less intuitive. TheDirichlet function, for example, is periodic, with any nonzero rational number serving as a period. However, it does not possess a fundamental period.

Complex-valued functions

Functions with a domain in thecomplex numbers can exhibit more complex periodic properties.

Complex exponential

The complex exponential function is a periodic function with a purely imaginary period:

e^{ikx}=\cos kx+i\,\sin kx

Given that the cosine and sine functions are both periodic with period $2\pi$ ,Euler's formula demonstrates that the complex exponential function has a period $L {\displaystyle L}$ such that

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

.

Double-periodic functions

A function on the complex plane can have two distinct, incommensurate periods without being a constant function. Theelliptic functions are a primary example of such functions. ("Incommensurate" in this context refers to periods that are not real multiples of each other.)

Properties

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 $n {\displaystyle n}$ ,^[3]

f(x+nP)=f(x)

A significant property related to integration is that if $f(x)$ is anintegrable periodic function with period $P {\displaystyle P}$ , then its definite integral over any interval of length $P {\displaystyle P}$ is the same.^[3] That is, for any real number $a {\displaystyle a}$ :

\int _{a}^{a+P}f(x)\,dx=\int _{0}^{P}f(x)\,dx

This property is crucial in areas such asFourier series, where the coefficients are determined by integrals over one period.

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 ${\frac {P}{|a|}}$ . For example, $f(x)=\sin(x)$ has period $2\pi$ and, therefore, $\sin(5x)$ will have period ${\frac {2\pi }{5}}$ .

A key property of many periodic functions is that they can be described by aFourier series. This series represents a periodic function as a sum of simpler periodic functions, namelysines and cosines. For example, a sound wave from a musical instrument can be broken down into the fundamental note and variousovertones. This decomposition is a powerful tool in fields like physics and signal processing. While most "well-behaved" periodic functions can be represented this way,^[4] Fourier series can only be used for periodic functions or for functions defined on a finite length. 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 anintegral over an interval of length $P {\displaystyle P}$ .

Any function that is a combination of periodic functions with the same period is also periodic (though its fundamental period may be smaller). This includes:

addition,subtraction, multiplication and division of periodic functions,^[1] and
taking a power or a root of a periodic function (provided it is defined for all $x {\displaystyle x}$ )

Generalizations

The concept of periodicity can be generalized beyond functions on the real number line. For example, the idea of a repeating pattern can be applied to shapes in multiple dimensions, such as a periodictessellation of the plane. Asequence can also be viewed as a function defined on thenatural numbers, and the concept of aperiodic sequence is defined accordingly.

Antiperiodic functions

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.^[5]

Bloch-periodic functions

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

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

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.^[6]

See also

Almost periodic function – Function that "converges" to periodicity
Amplitude – Measure of change in a periodic variable
Continuous wave – Electromagnetic wave that is not pulsed
Definite pitch – Easily discernible pitch
Double Fourier sphere method – Mathematical technique
Doubly periodic function – Function with two complex number "periods"
Fourier transform for computing periodicity in evenly spaced data
Frequency – Number of occurrences or cycles per unit time
Hill differential equation – Second order linear differential equation featuring a periodic function
Least-squares spectral analysis for computing periodicity in unevenly spaced data
List of periodic functions
Periodic sequence – Sequence for which the same terms are repeated over and over
Periodic summation – Sum of a function's values every _P_ offsets
Periodic travelling wave – Constant speed wavetrain
Quasiperiodic function – Class of functions behaving "like" periodic functions
Seasonality – Variations in data at specific regular intervals less than a year
Secular variation – Long-term non-periodic variation
Spectrum (physical sciences) – Concept relating to waves and signals
Wavelength – Distance over which a wave's shape repeats

References

^^a ^bTolstov, Georgij Pavlovič; Tolstov, Georgij Pavlovič (2009).Fourier series. Dover books on mathematics (Nachdr. ed.). New York: Dover Publ. p. 1.ISBN 978-0-486-63317-6.
^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 periods $P {\displaystyle P}$ being zero).
^^a ^bTolstov, Georgij Pavlovič (2009).Fourier series. Dover books on mathematics (Nachdr. ed.). New York: Dover Publ. p. 2.ISBN 978-0-486-63317-6.
^For instance, forL² functions,Carleson's theorem states that they have apointwise (Lebesgue)almost everywhere convergent Fourier series.
^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.

External links

"Periodic function".Encyclopedia of Mathematics.EMS Press. 2001 [1994].
Weisstein, Eric W."Periodic Function".MathWorld.

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