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 ?
N'Gueereekata, Almost Automorphic Functions and Almost Periodic Functions in Abstract Spaces, Kluwer Academic/Plenum Publishers, New York, London, Moscow, 2001.
I., Classes of periodic functions and approximation of their elements by Fourier sums, Dokl.
However, in our situation, p(t) is a nonconstant periodic function and, by denoting its fundamental period by T, we can choose the intervals [I.sub.i] such that [[theta].sup.[+ or -].sub.i+2] = [[theta].sup.[+ or -].sub.i] + T for all i [member of] Z, without any further hypothesis.
Corduneanu, A scale of almost periodic function spaces, Differential and Integral Equations, 24 (1-2) (2011), 1-27.
Since the smooth periodic function Fp can be approximated well by the partial sum of its Fourier series [5,7,10], from this inequality, we see that we have constructed a trigonometric polynomial [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which can approximate to f on [OMEGA] very well.
A periodic function is such that a given interval of length p, where p is the period, repeats its graph over and over again throughout the time line.
Sylvester and Cayley first showed that the function can be written in the form A(t) + U(t), where A(t) is a polynomial in t of degree N and U (t) is a periodic function of period the least common multiple of [a.sub.1], ..., [a.sub.r](see [5, 6] and references therein).
It is probably impossible to generate a uniform approximation within the limits of a linear model in the whole interval [0; 40] because periodic functions [f.sub.I], [f.sub.II] in intervals [0; 15] and [15; 40] (this follows from expression (3.9) differ not only in the amplitude but also in the period.
We improve this estimate by deriving a much simpler expression for the periodic function, including its average value [C.sub.h].
The ROM LUT approach uses sampled values of a periodic function stored in ROM.
Then for each k [greater than or equal to] 1 the function the [b.sub.k](u) is a periodic function with period 1.
It represents the situation where a linear trend takes the lead over a long-range stationarity of a periodic function for large values of [tau].

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