A function f is said to be periodic if, for some nonzero constant P, it is the case that
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 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.
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.
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.
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.)
Any function that consists only of periodic functions with the same period is also periodic (with period equal or smaller), including:
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:
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:
Consider a real waveform consisting of superimposed frequencies, expressed in a set as ratios to a fundamental frequency, f: F = 1⁄f [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 = LCD⁄f. Consider that for a simple sinusoid, T = 1⁄f. Therefore, the LCD can be seen as a periodicity multiplier.
If no least common denominator exists, for instance if one of the above elements were irrational, then the wave would not be periodic.