Old Web

English

Sign In

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 that is not periodic is called aperiodic. 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 that is not periodic is called aperiodic. 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. 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, 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. For example, the sine function is periodic with period 2 π {displaystyle 2pi } , since for all values of x {displaystyle x} . This function repeats on intervals of length 2 π {displaystyle 2pi } (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 f {displaystyle f} that gives the 'fractional part' of its argument. Its period is 1. In particular,

Parent Topic

Child Topic

No Parent Topic