5.2 Periodic functions

In physical world around us, we encounter many phenomena which repeat after certain interval of time. In mathematics, the notion of periodicity remains same but with more general connotation. The periodicity of a function is not limited to time. We look for repetition of function values with respect to independent variable. Time could be just one such independent variable. For example, we have seen that trigonometric functions are “many one” relations. This means that we get same value of trigonometric function for different angles. This “many one” relation is the basic requirement for a function to be periodic. In addition, these same values of the function should appear at regular intervals for the values of independent variables in the domain.

We can visualize periodic nature of a function by observing its graph in which a particular smallest segment of the plot can be repeated to construct complete plot.

In the two graphs shown above, we have considered two segments corresponding to intervals of “ $\pi$ ” and “ $2\pi$ ”. The graphs are constructed by repeating the segments one after another. It is clear from the figure that we need smallest segment of an interval “ $2\pi$ ” to construct a sine curve (lower curve).

Definition of periodic function

A function is said to be periodic if there exists a positive real number “T” such that

$$f\left(x+T\right)=f\left(x\right)\text{for all}x\in D$$

where “D” is the domain of the function f(x). The least positive real number “T” (T>0) is known as the fundamental period or simply the period of the function. The “T” is not a unique positive number. All integral multiple of “T” within the domain of the function is also the period of the function. Hence,

$$f\left(x+nT\right)=f\left(x\right);n\in Z,\text{for all}x\in D$$

In the context of periodic function, an “aperiodic” function is one, which in not periodic. On the other hand, a function is said to be anti-periodic if :

$$f\left(x+T\right)=-f\left(x\right)\text{for all}x\in D$$

Periodicity and period

In order to determine periodicity and period of a function, we can follow the algorithm as :

• Put f(x+T) = f(x).
• If there exists a positive number “T” satisfying equation in “1” and it is independent of “x”, then f(x) is periodic. Otherwise, function, “f(x)” is aperiodic.
• The least value of “T” is the period of the periodic function.