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This is a list of some well-known periodic functions. The constant function f ( x ) = c , where c is independent of x , is periodic with any period, but lacks a fundamental period . A definition is given for some of the following functions, though each function may have many equivalent definitions.
For example, the trigonometric functions, which repeat at intervals of radians, are periodic functions. 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.
Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials. Natural logarithm; Common logarithm; Binary logarithm; Power functions: raise a variable number to a fixed power; also known as Allometric functions; note: if the power is a rational number it is not strictly a transcendental function. Periodic ...
The doubly periodic function is thus a two-dimensional extension of the simpler singly periodic function, which repeats itself in a single dimension. Familiar examples of functions with a single period on the real number line include the trigonometric functions like cosine and sine , In the complex plane the exponential function e z is a singly ...
The Dirichlet function is therefore an example of a real periodic function which is not constant but whose set of periods, the set of rational numbers, is a dense subset of . Integration properties [ edit ]
For example, together with the uniform boundedness principle, ... The Dirichlet kernel is a periodic function which becomes the Dirac comb, ...
In those days the theory of elliptic functions and the theory of doubly periodic functions were considered to be different theories. They were brought together by Briot and Bouquet in 1856. [20] Gauss discovered many of the properties of elliptic functions 30 years earlier but never published anything on the subject. [21]
The graph of the Dirac comb function is an infinite series of Dirac delta functions spaced at intervals of T. In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function with the formula := = for some given period . [1]