Periodic Function

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periodic function

[¦pir·ē¦äd·ik ¦fəŋk·shən]
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)


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 ?
2006, Pseudo almost periodic solution to some semilinear differential equations, Math.
Because of the coupling of the force terms in (7) via the relation given by (5), the periodic solution set [q.
In Section 3, formulas for determining the direction of Hopf bifurcation and stability of periodic solutions are derived.
The task of finding of periodic solutions of nonlinear differential equations is more complex than the Cauchy task of integrating of these equations from the initial conditions to establishment because it imposes another condition on the solution--the condition of periodicity, that is it becomes a point-to-point T-periodic boundary value problem of nonlinear differential equations.
In the literatures, although there are numerous results on the existence and stability of neural networks with delays, the problem of global exponential stability of almost periodic solution for neutral-type Cohen-Grossberg SICNNs has not been fully investigated.
and there exists at least one positive periodic solution.
The stability of the periodic solution is determined by the sign of the eigenvalue of the Jacobian matrix of (451).
CHEBAN, Asymptotic Almost Periodic Solutions of Differential Equations, Hindawi Publishing Corporation, New York 2009.
0] depending on |g|, |h|, [nu], [rho] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then the periodic solution of (BF) satisfying [(#).
In the present paper, we obtain an approximate expression for the periodic solutions to two practical cases [25, 26] of nonlinear SODF oscillation systems, namely oscillation of current-carrying wire in a magnetic field and the model of bucking of a column by means of iteration perturbation method (IPM), variational approach (VA), and perturbation expansion method (PEM).