Periodic Functions

Function y=f(x) is called periodic, if exists such number `T!=0`, that for any x from domain of the function we have the following: `f(x+T)=f(x)=f(x-T)`.

Number T is called period of the function y=f(x).

From definition it follows that periodic functions has infinitely many periods. For example, if T is period of function, then any number of form kT, where k is integer number, is also period of function.

Often (but not always) among set of positive periods of the function we can find the smallest one. This period is called main period.

Usually, only main period is considered and it is called simply "period".

Examples of periodic functions (with specifying of main period):

  1. `y={x}` -period T=1; (see note function y={x})
  2. `y=sin(x)` -period `T=2pi`; (see note properties and graph of the function y=sin(x))
  3. `y=tan(x)` -period `T=pi`. (see note properties and graph of the function y=tan(x))

Fact. If function f is periodic and has period T, then function `Af(kx+b)` (where A,k and b are constant, `k!=0`) is also periodic with period `T/(|k|)`.

For example, period of the function `2sin(3x-pi/6)` is `(2pi)/3`; period of the function `tan(-x/3+pi/4)` is `pi/(|-1/3|)=3pi`.