5.2 Periodic functions  (Page 2/5)

Basic periodic functions

Not many of the functions that we encounter are periodic. There are few functions, which are periodic by their very definition. We are, so far, familiar with following periodic functions in this course :

• Constant function, (c)
• Trigonometric functions, (sinx, cosx, tanx etc.)
• Fraction part function, {x}

Six trigonometric functions are most commonly used periodic functions. They are used in various combination to generate other periodic functions. In general, we might not determine periodicity of each function by definition. It is more convenient to know periods of standard functions like that of six trigonometric functions, their integral exponents and certain other standard forms/ functions. Once, we know periods of standard functions, we use different rules, properties and results of periodic functions to determine periods of other functions, which are formed as composition or combination of standard periodic functions.

Constant function

For constant function to be periodic function,

$$f\left(x+T\right)=\mathrm{f(x)}$$

By definition of constant function,

$$f\left(x+T\right)=\mathrm{f(x)}=c$$

Clearly, constant function meets the requirement of a periodic function, but there is no definite, fixed or least period. The relation of periodicity, here, holds for any change in x. We, therefore, conclude that constant function is a periodic function without period.

Trigonometric functions

Graphs of trigonometric functions (as described in the module titled trigonometric function) clearly show that periods of sinx, cosx, cosecx and secx are “2π” and that of tanx and cotx are “π”. Here, we shall mathematically determine periods of few of these trigonometric functions, using definition of period.

Sine function

For sinx to be periodic function,

$$\sin \left(x+T\right)=\left(x\right)$$

$$x+T=n\pi +{\left(-1\right)}^{n}x;n\in Z$$

The term ${\left(-1\right)}^n$ evaluates to 1 if n is an even integer. In that case,

$$x+T=n\pi +x$$

Clearly, T = nπ, where n is an even integer. The least positive value of “T” i.e. period of the function is :

$$T=2\pi$$

Cosine function

For cosx to be periodic function,

$$\cos \left(x+T\right)=\cos x$$

$$\Rightarrow x+T=2n\pi \pm x;n\in Z$$

Either,

$$\Rightarrow x+T=2n\pi +x$$

$$\Rightarrow T=2n\pi$$

or,

$$\Rightarrow x+T=2n\pi -x$$

$$\Rightarrow T=2n\pi -2x$$

First set of values is independent of “x”. Hence,

$$T=2n\pi ;n\in Z$$

The least positive value of “T” i.e. period of the function is :

$$T=2\pi$$

Tangent function

For tanx to be periodic function,

$$\tan (x+T)=\tan x$$ $$x+T=n\pi +x;n\in Z$$

Clearly, T = nπ; n∈Z. The least positive value of “T” i.e. period of the function is :

$$T=\pi$$

Fraction part function (fpf)

Fraction part function (FPF) is related to real number "x" and greatest integer function (GIF) as $\left\{x\right\}=x-\left[x\right]$ . We have seen that greatest integer function returns the integer which is either equal to “x” or less than “x”. For understanding the nature of function, let us compute few function values as here :

--------------------------------- x [x]x – [x] ---------------------------------1 1 0 1.25 1 0.251.5 1 0.5 1.75 1 0.752 2 0 2.25 2 0.252.5 2 0.5 2.75 2 0.753 3 0 3.25 3 0.253.5 3 0.5 3.75 3 0.754 4 0 ---------------------------------