# Periodic Function

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## periodic function

[¦pir·ē¦äd·ik ¦fəŋk·shən]*x*) of a real or complex variable is periodic with period

*T*if ƒ(

*x*+

*T*) = ƒ(

*x*) for every value of

*x*.

## Periodic Function

a function whose value does not change when its argument is increased by a certain nonzero number called the period of the function. For example, sin *x* and cos *x* are periodic functions with period 2π; {*x*}—the fractional part of the number *x*—is a periodic function with period 1; and the exponential function *e ^{x}* (if

*x*is a complex variable) is a periodic function with period 2

*πi*.

The sum or difference of two periods is a period; consequently, any multiple of a period is also a period. It follows that every periodic function has an infinite set of periods. If a periodic function has a real period, is continuous, and is not constant, there exists a smallest positive period *T;* every other real period of the function is of the form *kT*, where *k* = ±1, ±2,…. The sum, product, and quotient of periodic functions with the same period are also periodic functions with that period. The derivative of a periodic function is a periodic function with the same period, but the integral of a periodic function *f* (*x*) with period *T* is a periodic function (with the same period) only if ∫_{0}* ^{T}f* (

*x*)

*dx*= 0. The fundamental theorem of the theory of periodic functions asserts that if a periodic function

*f*(

*x*) with period

*T*obeys certain conditions—such as that

*f*(

*x*) is continuous and has only a finite number of maxima and minima in the interval (0,

*T*)—it can be expressed as a convergent trigonometric series, or Fourier series, of the form

The coefficients of this series can be expressed in terms of *f* (*x*) through the Euler-Fourier formulas.

A continuous periodic function of a complex variable may have two periods *T*_{1} and *T*_{2}, whose ratio is not a real number: if the function is not constant, each of its periods has the form *k*_{1}*T*_{1} + *k*_{2}*T*_{2}, where *k*_{1} = 0, ±1, ±2, … and *k*_{2} = 0, ±1, ±2, … In this case the periodic function is said to be doubly periodic. We also speak of doubly periodic functions of the second and third kinds; these are functions that change, respectively, by a constant or exponential multiplier when their arguments are increased by their periods:

*f* (*x* + *T*_{1}) = *a*_{1}*f* (*x*) and *f* (*x* + *T*_{2}) = *a*_{2}*f* (*x*)

or

*f* (*x* + *T*_{1}) = *e ^{a}*

_{1}

*f*(

*x*) and

*f*(

*x*+

*T*

_{2}) =

*e*

^{a}_{2}

*f*(

*x*)

The sum of periodic functions with incommensurable periods is not a periodic function; for example, cos *x* + cos (*x*√2) is not a periodic function. Functions of this kind, however, have many properties in common with periodic functions and are the simplest examples of so-called almost periodic functions. Periodic functions play an extremely large role in the theory of oscillations and in mathematical physics in general.