# Periodic function

Last updated

A periodic function is a [function (mathematics)|function] that repeats its values at regular intervals, for example, the trigonometric functions, which repeat at intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.

## Definition

A function f is said to be periodic if, for some nonzero constant P, it is the case that

${\displaystyle f(x+P)=f(x)}$

for all values of x in the domain. A nonzero constant P for which this is the case is called a period of the function. If there exists a least positive [1] constant P 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 P will repeat on intervals of length P, 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 f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. 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 ${\displaystyle 2\pi }$, since

${\displaystyle \sin(x+2\pi )=\sin x}$

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

Everyday examples are seen when the variable is time; for instance the hands of a clock 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 ${\displaystyle f}$ that gives the "fractional part" of its argument. Its period is 1. In particular,

${\displaystyle f(0.5)=f(1.5)=f(2.5)=\cdots =0.5}$

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

The trigonometric functions sine and cosine are common periodic functions, with period 2π (see the figure on the right). The subject of Fourier 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 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:

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

Since the cosine and sine functions are both periodic with period 2π, the complex exponential is made up of cosine and sine waves. This means that Euler's formula (above) has the property such that if L is the period of the function, then

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

#### 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 ${\displaystyle f}$ is periodic with period ${\displaystyle P}$, then for all ${\displaystyle x}$ in the domain of ${\displaystyle f}$ and all positive integers ${\displaystyle n}$,

${\displaystyle f(x+nP)=f(x)}$

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

Some periodic functions can be described by Fourier series. For instance, for L2 functions, Carleson's theorem states that they have a pointwise (Lebesgue) almost everywhere convergent Fourier series. Fourier series can only be used for periodic functions, or for functions on a bounded (compact) interval. If ${\displaystyle f}$ is a periodic function with period ${\displaystyle P}$ that can be described by a Fourier series, the coefficients of the series can be described by an integral over an interval of length ${\displaystyle P}$.

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).

## Generalizations

### Antiperiodic functions

One common subset of periodic functions is that of antiperiodic functions. This is a function f such that f(x + P) = f(x) for all x. (Thus, a P-antiperiodic function is a 2P-periodic function.) For example, the sine and cosine functions are π-antiperiodic and 2π-periodic. While a P-antiperiodic function is a 2P-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, which govern the solution of various periodic differential equations. In this context, the solution (in one dimension) is typically a function of the form:

${\displaystyle f(x+P)=e^{ikP}f(x)}$

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 = π/P.

### Quotient spaces as domain

In signal processing you encounter the problem, that Fourier series represent periodic functions and that Fourier series satisfy convolution 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 a quotient space:

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

That is, each element in ${\displaystyle {\mathbb {R} /\mathbb {Z} }}$ is an equivalence class of real numbers that share the same fractional part. Thus a function like ${\displaystyle f:{\mathbb {R} /\mathbb {Z} }\to \mathbb {R} }$ 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 a fundamental frequency, f: F = 1f[f1 f2 f3 … fN] 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 = LCDf. Consider that for a simple sinusoid, T = 1f. Therefore, the LCD can be seen as a periodicity multiplier.

• For set representing all notes of Western major scale: [1 9854433253158] the LCD is 24 therefore T = 24f.
• For set representing all notes of a major triad: [1 5432] the LCD is 4 therefore T = 4f.
• For set representing all notes of a minor triad: [1 6532] the LCD is 10 therefore T = 10f.

If no least common denominator exists, for instance if one of the above elements were irrational, then the wave would not be periodic. [2]

## Related Research Articles

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

In mathematics, the trigonometric 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 sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.

In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle of the summation can be made to approximate an arbitrary function in that interval. As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.

In mathematics, the discrete sine transform (DST) is a Fourier-related transform similar to the discrete Fourier transform (DFT), but using a purely real matrix. It is equivalent to the imaginary parts of a DFT of roughly twice the length, operating on real data with odd symmetry, where in some variants the input and/or output data are shifted by half a sample.

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

A sine wave or sinusoid is a mathematical curve that describes a smooth periodic oscillation. A sine wave is a continuous wave. It is named after the function sine, of which it is the graph. It occurs often in both pure and applied mathematics, as well as physics, engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

In mathematics, the Gibbs phenomenon, discovered by Henry Wilbraham (1848) and rediscovered by J. Willard Gibbs (1899), is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as n increases, but approaches a finite limit. This sort of behavior was also observed by experimental physicists, but was believed to be due to imperfections in the measuring apparatus.

In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as p-functions and they are usually denoted by the symbol ℘. They play an important role in theory of elliptic functions. A ℘-function together with its derivative can be used to parameterize elliptic curves and they generate the field of elliptic functions with respect to a given period lattice.

In mathematics and in signal processing, the Hilbert transform is a specific linear operator that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). This linear operator is given by convolution with the function . The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° to every frequency component of a function, the sign of the shift depending on the sign of the frequency. The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal u(t). The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.

In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function is an even function if n is an even integer, and it is an odd function if n is an odd integer.

In mathematics, a rose or rhodonea curve is a sinusoid specified by either the cosine or sine functions with no phase angle that is plotted in polar coordinates. Rose curves or "rhodonea" were named by the Italian mathematician who studied them, Guido Grandi, between the years 1723 and 1728.

In mathematics, trigonometric interpolation is interpolation with trigonometric polynomials. Interpolation is the process of finding a function which goes through some given data points. For trigonometric interpolation, this function has to be a trigonometric polynomial, that is, a sum of sines and cosines of given periods. This form is especially suited for interpolation of periodic functions.

In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur.

In mathematics, the sine is a trigonometric function of an angle. The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle. For an angle , the sine function is denoted simply as .

In mathematics, particularly the field of calculus and Fourier analysis, the Fourier sine and cosine series are two mathematical series named after Joseph Fourier.

In mathematical analysis, the Dirichlet kernel, named after the German mathematician Peter Gustav Lejeune Dirichlet, is the collection of functions defined as

In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin in the Cartesian coordinate system in the Euclidean plane. In topology, it is often denoted as S1 because it is a one-dimensional unit n-sphere.

## References

1. For some functions, like a constant function or the Dirichlet function (the indicator function of the rational numbers), a least positive period may not exist (the infimum of all positive periods P being zero).
• Ekeland, Ivar (1990). "One". Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. 19. Berlin: Springer-Verlag. pp. x+247. ISBN   3-540-50613-6. MR   1051888.