Periodic Function

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periodic function

[¦pir·ē¦äd·ik ¦fəŋk·shən]
(mathematics)
A function ƒ(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 ex (if x is a complex variable) is a periodic function with period 2πi.

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 ∫0Tf (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 T1 and T2, whose ratio is not a real number: if the function is not constant, each of its periods has the form k1T1 + k2T2, where k1 = 0, ±1, ±2, … and k2 = 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 + T1) = a1f (x) and f (x + T2) = a2f (x)

or

f (x + T1) = ea1f (x) and f (x + T2) = ea2f (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.

References in periodicals archive ?
The 1-year periodic signal of the BU case series was associated with Nyong River flow from the end of 2005 to the end of 2009 (Figure 1) and with rainfall from the end of 2005 to the beginning of 2011 (online Technical Appendix Figure 3).
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