Periodic function

In Mathématiques, a periodic function is a function which when it is applied to a variable, takes again the same value if one adds to this variable a certain fixed quantity called period . Examples of such functions can be obtained starting from periodic phenomena, as the hour indicated by the small needle of a Horloge, the Phases of the moon, etc

Definition

A definite function F a unit

\ mathcal D

of real numbers or integers, is periodic of period T or T - periodic if

\ forall X \ in \ mathcal D, x+t \ in \ mathcal D

and

\ forall X \ in \ mathcal D, F (x+t) =f (X) \

For a periodic function, the graph can be traced by recopying in a repetitive way, a particular portion length one period, with regular intervals: it is a property of Symétrie by translation.

For example, the fractional function left F which associates with a real number its fractional part defined by

\ forall X \ in \ mathbb {R}, F (X) =x-

(being the whole Left X ) is periodic and of period 1. Thus we have

F (0,5) = F (1,5) = F (2,5) =… = 0,5.

If a function F is periodic of period T then for all X pertaining to the whole of definition of F and for entire N

F ( X + NT ) = F ( X ).

In the preceding example, the function being of period 1, we have for any reality X

F (X) = F (X + 1) = F (X + 2)…

The functions sine and cosine are periodic and of period 2π.

The theory of the Fourier series seeks to write an arbitrary periodic function like a sum of goniometrical functions.

In Physical, a periodic movement is a movement in which the position, (or positions) of a system is exprimables using periodic functions of time, having all the same period. See for example the articles Onde in tooth of saw, and triangular Onde

Average, derived and primitive from the numerical periodic functions

Median value

The value Moyenne of a periodic function F integrable of period T is the following value, which is independent of driven a:

\ = \ frac {1} {T} {\ int_0^T F (X) \, dx} = \ frac {1} {T} {\ int_a^ {a+T} F (X) \, dx}

Thus the function cosine is of null average, its square of average 1/2.

Even if it means to add a constant to the function, one can change his median value.

Derived and primitive

the derivative of a function $\ mathcal C^1$ , T-periodical, is T periodic and of null average
a function F continues and T periodic admits a periodic primitive T if and only if F is of null average (all the primitives are then periodic, only one being of null average).

For a more precise study of the properties of derivation for the periodic functions, it is necessary to introduce the Fourier series; one can then show the Inégalité of Wirtinger which compares the standards of F and its derivative.

More general definition and together of periods

Are E a unit provided with a Law of composition additivement interns noted + and F an unspecified unit. An element T of E being given, a function T-periodical or function of period T of E in F , is a function of E in F such as:

\ forall X \ in E, F (x+T) =f (X)

Let us notice that unless the law is supposed commutative this definition depends on the place of T on the right.

More frequently, E is a commutative Groupe whose whole of the possible periods forms a sub-group. If one then considers the whole of the strictly positive periods of a function F, if F is continuous and that this unit then does not admit a Plus small element F is constant, if not F admits a smaller strictly positive period .