2.18 Periodic Functions

In physics and engineering we see periodic phenomena such as vibrations, the motion of the tide, planetary, and alternating current (AC).

We say a function f is periodic with period T if x+T lies in the domain of f whenever x lies in the domain of f and if for every x in the domain of f
f(x+T)=f(x)
  •  The condition “x+T lies in the domain of f whenever x lies in the domain of f” does not say that f needs to be defined for all x. For example,
  •  The period of a periodic function is not unqiue. In fact, if f is periodic with T, it is also periodic with periods 2T, 3T,… , or −T and −2T,… because
    f(x+2T)=f((x+T)+T)=f(x+T)=f(x) and
    f(x−T)=f(x−T+T)=f(x), and so on. In general:
If f is periodic with period T then
f(x+nT)=f(x) for every integer n.

 

The smallest positive period of a periodic function (if exists) is called its fundamental period.

 

  • A constant function f(x)≡3 is periodic, and every number T is its period
    f(x+T)=f(x)=3. Therefore, there is no smallest period, and hence f does not have a fundamental period.
Example84
Determine the fundamental period of f(x)=2x−⌊2x⌋.
Solution
Let T be the period of f. Thus
f(x+T)=f(x)2(x+T)−⌊2(x+T)⌋=2x−⌊2x⌋2x+2T−⌊2(x+T)⌋=2x−⌊2x⌋
Therefore
2T=⌊2(x+T)⌋−⌊2x⌋ Because both ⌊2(x+T)⌋ and ⌊2x⌋ are integers,2T must be an integer too. To determine the fundamental period,we choose 2T to be the smallest positive integer, 1:
2T=1⇒T=12. That is, the fundamental period of f is 1/2. The graph of this
function is shown below.
Example85
Show that f(x)=x2−⌊x2⌋ is not periodic.
Solution
Assume f is periodic with period T. Then
f(x+T)=f(x) ⇒(x+T)2−⌊(x+T)2⌋=x2−⌊x2⌋ Expanding (x+T)2, we get
x2+2xT+T2−⌊(x+T)2⌋=x2−⌊x2⌋ or
2xT+T2=⌊(x+T)2⌋−⌊x2⌋ The right hand side is always an integer, so the right hand side for every real nymber x must be an integer too, and that is impossible. {[}For any number T, 2xT+T is a linear function that can take any real value not only integers{]}. The graph of this function is shown below.