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In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
Examples
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. The massive clock on the Clock Tower of the Palace of Westminster, London (commonly known as Big Ben, although Big Ben is the bell inside - the picture is St Stephens Tower). ...
Apparent magnitude: up to -12. ...
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. More explicitly, a function f is periodic with period P greater than zero if In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...
The integers are commonly denoted by the above symbol. ...
- f(x + P) = f(x)
for all values of x in the domain of f. An aperiodic function (non-periodic function) is one that has no such period P (not to be confused with an antiperiodic function, below, for which f(x + P) = −f(x) for some P).
If a function f is periodic with period P then for all x in the domain of f and all integers n,
- f( x + Pn ) = f ( x ).
In the above example, the value of P is 1, since f( x ) = f( x + 1 ) = f( x + 2 ) = etc. The period of a function need not be the smallest value (least period) that satisfies the above equation, so P could also equal two.
 A plot of the f( x) = sin( x) and f( x) = cos( x) functions, both with period 2Pi. A simple example is the function f that gives the "fractional part" of its argument: Image File history File links Sine_cosine_plot. ...
Image File history File links Sine_cosine_plot. ...
- f( 0.5 ) = f( 1.5 ) = f( 2.5 ) = ... = 0.5.
Some named examples are sawtooth wave, square wave and triangle wave. This article or section does not cite its references or sources. ...
A square wave is a kind of basic waveform. ...
A triangle wave is a waveform named for its triangular shape. ...
The trigonometric functions sine and cosine are common periodic functions, with period 2π. The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods. All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometric functions: , , , , , In mathematics, the trigonometric functions (also called circular functions) are functions of an angle; they are important when studying triangles and modeling periodic phenomena, among many other...
The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...
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.) In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ...
General definition Let E be a set with an internal operation + . A T-periodic function, or function periodic with period T on E is a function f on E to some set F, such that In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as a subset and possesses some given property. ...
In logic and mathematics, an operation ω is a function of the form ω : X1 × … × Xk → Y. The sets Xj are the called the domains of the operation, the set Y is called the codomain of the operation, and the fixed non-negative integer k is called the arity of the operation. ...
Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A...
- for all x in E, f(x + T) = f(x).
Note that unless + is assumed commutative this definition depends on writing T on the right. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ...
The period T is not unique. For a given T, every integer multiple of T is also a period.
Antiperiodic functions and other generalizations One common generalization of periodic functions is that of antiperiodic functions. This is a function f such that f(x + T) = −f(x) for all x. (Thus, a T-antiperiodic function is a 2T-periodic function.)
A further generalization appears in the context of Bloch waves 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: A Bloch wave or Bloch state is the wavefunction of a particle (usually, an electron) placed in a periodic potential. ...
Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to linear differential equations of the form, , with a continuous periodic function with period . ...
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/T.
Periodic sequences Some naturally-occurring sequences are periodic, for example (eventually) the decimal expansion of any rational number (see recurring decimal). We can therefore speak of the period or period length of a sequence. This is (if one insists) just a special case of the general definition. In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
The decimal (base ten or occasionally denary) numeral system has ten as its base. ...
In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ...
A recurring or repeating decimal is a number which when expressed as a decimal has a set of final digits which repeat an infinite number of times. ...
Translational symmetry If a function is used to describe an object, e.g. an infinite image is given by the color as function of position, the periodicity of the function corresponds to translational symmetry of the object. Sphere symmetry group o. ...
See also In mathematics, almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. ...
Amplitude is a nonnegative scalar measure of a waves magnitude of oscillation, that is, magnitude of the maximum disturbance in the medium during one wave cycle. ...
In music a sound or note of definite pitch is one of which it is possible or relatively easy to discern the pitch or frequency of the fundamental, as opposed to sounds of indefinite pitch. ...
FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ...
Oscillation is the variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ...
In mathematics, almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. ...
The wavelength is the distance between repeating units of a wave pattern. ...
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