In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic.
A function f is said to be periodic with period P (P being a nonzero constant) if we have
for all values of x in the domain. If there exists a least positive constant P with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) A function with period P will repeat on intervals of length P, and these intervals are referred to as periods.
Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, 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 periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.
A function that is not periodic is called aperiodic.
For example, the sine function is periodic with period 2π, since
for all values of x. This function repeats on intervals of length 2π (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.
A simple example of a periodic function is the function f that gives the "fractional part" of its argument. Its period is 1. In particular,
- f( 0.5 ) = f( 1.5 ) = f( 2.5 ) = ... = 0.5.
The graph of the function 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.
If a function f is periodic with period P, then for all x in the domain of f and all integers n,
- f(x + nP) = f(x).
If f(x) is a function with period P, then f(ax+b), where a is a positive constant, is periodic with period P/|a|. For example, f(x)=sinx has period 2π, therefore sin(5x) will have period 2π/5.
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.)
Using complex variables we have the common period function:
As you can see, since the cosine and sine functions are periodic, and the complex exponential above is made up of cosine/sine waves, then the above (actually Euler's formula) has the following property. If L is the period of the function then:
One common generalization 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 or cosine function is π-antiperiodic and 2π-periodic.
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:
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:
- 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.
- Hazewinkel, Michiel, ed. (2001), "Periodic function", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Periodic functions at MathWorld