<< Chapter < Page Chapter >> Page >

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.

Periodic function

A sine curve is constructed by repeating a segment of the curve as shown in the lower graph in the figure.

In the two graphs shown above, we have considered two segments corresponding to intervals of “ π ” and “ 2 π ”. 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 π ” 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 x + T = f x for all x 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 x + n T = f x ; n Z , for all x 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 x + T = - f x for all x 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.

Problem : Let f(x) be a function and “k” be a positive real number such that :

f x + k + f x = 0 for all x R

Prove that f(x) is periodic. Also determine its period.

Solution : The given equation can be re-written as :

f x + k = - f x for all x R

Here, our objective is to convert RHS of the equation as f(x). For this, we need to substitute "x" such that RHS function acquires RHS function form. Replacing “x” by “x+k”, we have :

f x + 2 k = - f x + k for all x R

Combining two equations,

f x + 2 k = - 1 X - f x = f x for all x R

It means that f(x) is a periodic function and its period is “2k”.

Problem : Determine period of the function :

f x = a sin k x + b cos k x

Solution : The function is sum of two trigonometric functions. We can reduce this function is terms of a single trigonometric function to determine its periodic nature. Let

a = r cos θ ; b = r sin θ

r = a 2 + b 2

Substituting in the function, we have :

f x = r cos θ sin k x + r sin θ cos k x = r sin k x + θ

This is a periodic function. Also, period of “ag(x)” is same as that of “g(x)”. Therefore, period of “r sin (kx + θ)” is same as that of “sin (kx + θ)”. On the other hand, period of g(ax+b) is equal to the period of g(x), divided by “|a|”. Now, period of “sinx” is “2π”. Hence, period of the given function is :

T = 2 π | k |

Alternatively, we can treat given function as addition of two functions. The period of each term is “2π/|k|”. Applying LCM rule (discussed later), the period of given function is equal to LCM of two periods, which is “2π/|k|”.

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Functions' conversation and receive update notifications?

Ask