Periodic Function & Fouries Series

Fourier Series:
Fourier series is an infinite series that is represented in terms of the trigonometric sine & cosine functions.

Most of the single valued function which occur in applied mathematics can be expressed in the form of Fourier series.

This is a powerful method to solve ordinary & partial differential equations particularly with periodic functions appearing as non homogeneous terms.

The application of Fourier Series are

• Electrical Engineering
• Vibration Analysis
• Acoustics
• Optics
• Signal Processing
• Image Processing
• Quantum Mechanics
• Econometrics
• Thin Walled Shell Theory

Taylor series expansion is valid only for functions which are continuous & differentiable. But Fourier series is possible not only for continuous functions but also for periodic functions which are discontinuous in their values & derivative.

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Periodic Function:

A function f(x) is said to be periodic if ∃ a number s.t. T >0,

f(x+T)=f(x)

where T is called period of f(x).

The periodic function show repetitive behavior that is it repeats itself in equal intervals & can be defined on a finite interval.

sinx, cosx, secx, cosecx are periodic with period 2π.

cotx, tanx are periodic with a period of π.

• sinx=sin(x+2π)=sin(x+4π)=.....
• sin3x=sin3(x+2π/3), period is 2π/3.
• cos5x=cos5(x+2π/5), period is 2π/5.

In general period of sinnx, cosnx= 2π/n.

Ex-1: Find the period of f(x)=2sinx+4cosx ?

 Sol: sinx & cosx are both periodic with a period of 2π.

f(x+2π)=2sin(x+2π)+4cos(x+2π)

            =2sinx+4cosx

            =f(x)

Ex-2: Find the period of f(x)=sinπx/l ?

Sol: Period=2π/n

                  = 2π/(π/l)

                  =2l.

Ex-3: Find the period of f(x)=1/3sin(4x+3) ?

Sol: Period=2π/n

                  = 2π/4

                  =π/2.

Ex-4: Find the period of f(x)=cot3x ?

Sol: Period=π/n

                  = π/3.

Ex-5: Find the period of f(x)=2sinx+3sin2x+1/3sin3x+4sin4x?

Sol: Period of sinx=2π

        Period of sin2x=π

        Period of sin3x=2π/3

        Period of sin4x=2π/4=π/2

So now period of f(x)=LCM of Numerator/HCF of Denominator 

                                   = LCM (2π, π, 2π, π)/HCF (1,1,3,2)

                                   =2π/1

                                   =2π.

Note:

• LCM rule is not always true. We can't apply this rule when functions are co-functions of each other or even function .
• If individual periods are rational or irrational number, LCM rule is not defined.

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