Chapter: Mathematics (maths) - Fourier Series

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Fourier Series

1 Introduction 2 Periodic Functions 3 Even And Odd Functions 4 Half Range Series 5 Parseval’s Theorem 6 Change Of Interval 7 Harmonic Analysis 8 Complex Form Of Fourier Series 9 Summary

FOURIER SERIES

 

1 INTRODUCTION

2 PERIODIC FUNCTIONS

3 EVEN AND ODD FUNCTIONS

4 HALF RANGE SERIES

5 PARSEVAL’S THEOREM

6 CHANGE OF INTERVAL

7 HARMONIC ANALYSIS

8 COMPLEX FORM OF FOURIER SERIES

9 SUMMARY

 

 

 

1 INTRODUCTION

 

The concept of Fourier series was first introduced by Jacques Fourier (1768– 1830), French Physicist and Mathematician. These series became a most important tool in Mathematical physics and had deep influence on the further development of mathematics it self.Fourier series are series of cosines and sines and arise in representing general periodic functions that occurs in many Science and Engineering problems. Since the periodic functions are often complicated, it is necessary to express these in terms of the simple periodic functions of sine and cosine. They play an important role in solving ordinary and partial differential equations.

 

2 PERIODIC FUNCTIONS

 

A function f (x) is called periodic if it is defined for all real „x‟ some positive number „p‟ such that

 

f (x + p ) = f (x) for all x.

This         number   „p‟   is   called   a   period   of   f(x

 

If a periodic function f (x) has a smallest period p (>0), this is often called the fundamental period of f(x). For example, the functions cosx and sinx have fundamental period 2p.

 

DIRICHLET CONDITIONS

 

Any function f(x), defined in the interval c ££c + 2p, can be developed as a Fourier series of the form


provided the following conditions are satisfied.

 

f (x) is periodic, single–valued and finite in [ c, c + 2 p]. 

f (x) has a finite number of discontinuities in [ c, c + 2p].

 

f (x) has at the most a finite number of maxima and minima in [ c,c+ 2p].

 

These conditions are known as Dirichlet conditions. When these conditions are satisfied, the Fourier series converges to f(x) at every point of continuity. At a point of discontinuity x = c, the sum of the series is given by

 

f(x) = (1/2) [ f (c-0) + f (c+0)] ,

where f (c-0) is the limit on the left and f (c+0) is the limit on the right.

 

EULER’S  FORMULAE

The Fourier series for the function f(x) in the interval c < x < c + 2pis given by


These values of a0, an, bn are   known   as   Euler‟s 0 , formula en,b The coefficients a0, an, bn  are  are also termed as Fourier coefficients.


 

Example 1

Expand f(x) = x as Fourier Series (Fs)  in the interval [ -π,   π]



 

Example 2

Expand   f(x) = xin the interval ( -£p) and hence deduce that






Example 3

Obtain the Fourier Series of periodicity   2πin [for-π, π]






 

Example 4




 


Example 5

Find the Fourier series for f (x) = (x + x2) in (-p< x < p) of percodicity 2and hence deduce that






Here x = -pand x = pare the end points of the range. \ The value of FS at x = pis the average of the values of f(x) at x = pand x = -p.







Exercises:

 

Determine the Fourier expressions of the following functions in the given interval

 

1.f(x) = (p- x)2, 0 < x < 2p 

2.f(x) = 0 in -p< x < 0

 

pin 0 < x < p 

3.f(x) = x –x2 in [-p,p

4.f(x) = x(2p-x) in (0,2p

5.f(x) = sinh ax in [-pp

6.f(x) = cosh ax in [-pp

7.f(x) = 1 in 0 < x < p

 

= 2 in p< x < 2p 

8.f(x) = -p/4 when -p< x < 0

 

=  p/4 when 0 < x < p

 

9.f(x) = cosax, in -p< x < p,   wherea‟is„ not   an   integer

 

10.Obtain a fourier series to represent e-ax from x = -pto x = p. Hence derive the series for p/ sinhp

 

 

3 Even and Odd functions

 

A function f(x) is said to be even if f (-x) = f (x). For example x2, cosx, x sinx, secx are even functions. A function f (x) is said to be odd if f (-x) = - f (x). For example, x3, sin x, x cos x,. are odd functions.


(1) The Euler‟s formula for even function is a


(2) The   Euler‟s   formula   for   odd   function   is


 

Example 6

Find the Fourier Series for f (x) = x in ( -pp)

Here, f(x) = x        is an odd function.


 

Example 7

Expand f (x) = |x| in (-pp) as FS and hence deduce that



 


 

Example 8


Then find the FS for f(x) and hence show that


Here f (-x) in (-p,0) = f (x) in (0,p)

 

f  (-x) in (0,p) = f (x) in (-p,0) 

 

f(x) is a even function




 

Example 9

 

Obtain the FS expansion of f(x) = x sinx in (-p< x<p) and hence deduce that


Here f (x) = xsinx  is an even function.






Exercises:

Determine Fourier expressions of the following functions in the given interval:


i. f(x) = p/2 + x, -p< x < 0

p/2 - x, 0 < x < p

ii. f(x) = -x+1 for + -p< x < 0

x+1 for 0 =< x =< p


 

 

4 HALF RANGE SERIES

 

It is often necessary to obtain a Fourier expansion of a function for the range (0, p) which is half the period of the Fourier series, the Fourier expansion of such a function consists a cosine or sine terms only.

 

(i) Half Range Cosine Series

 

The Fourier cosine series for f(x) in the interval (0,p) is given by


 

(ii)    Half Range Sine Series 

 

The Fourier sine series for f(x) in the interval (0,p) is given by




Example 10

 

If  c is the constant in ( 0 < x < p) then show that

 

c = (4c / p)  { sinx + (sin3x /3) + sin5x / 5) + ... ... ... }

Given   f(x) = c in (0,p).



 

Example 11

Find the Fourier Half Range Sine Series and Cosine Series for f(x) = x in the interval (0,p).


Sine Series






 

Example 12

 

Find the sine and cosine half-range series for the function function . f(x) = x , 0 < x £π/2

 

= π-x,  £x< pπ/2

 

Sine series




 




Exercises

 

1.Obtain cosine and sine series for f(x) = x in the interval 0< x < p. Hence show that 1/12


2.Find the half range cosine and sine series for f(x) = x2 in the range 0 < x < p

 

3.Obtain the half-range cosine series for the function f(x) = xsinx in (0,p)..

 

4.Obtain cosine and sine series for f(x) = x (p-x) in 0< x < p 5.Find the half-range cosine series for the function

 

6.f(x) = (px) / 4 , 0<x< (p/2)

 

= (p/4)(p-x), p/2 < x < p.

 

7.Find half range sine series and cosine series for

 

f(x) = x in  0<x< (p/2)

 

= 0 in  p/2 < x < p.

 

8.Find half range sine series and cosine series for the function f(x) == p- x in the interval 0 < x < p.

9.Find the half range sine series of f(x) = x cosx in (0,p)

 

10.Obtain cosine series for

 

f(x) = cos x ,   0<x< (p/2)

 

=   0,  p/2 < x < p

 

 

 

5 Parseval’s Theorem 

 

Root Mean square value of the function f(x) over an interval (a, b) is defined as


 

The use  of r.m.s value of a periodic  function is frequently made in the theory of mechanical vibrations and in electric circuit theory. The r.m.s value  is known as the effective value of the  function.

 

Parseval’s   Theorem

If f(x) defined in the interval (c, c+2π ), then the Parseval‟s Identity is given by 


 

Example 13

Obtain the Fourier series for f(x) 



 


 

CHANGE OF INTERVAL

 

In most of the Engineering applications, we require an expansion of a given function over an interval 2l other than 2p.

 

Suppose f(x) is a function defined in the interval c< x < c+2l. The Fourier expansion for f(x) in the interval c<x<c+2l is given by

 


 

 

Even and Odd Function

 

If f(x) is an even function and is defined in the interval ( c, c+2 l ), then 


 

Half Range Series

Sine Series



Cosine series


 

 

Example 14

Find the Fourier series expansion for the function



 

 

Example 15

 

Find the Fourier series of periodicity 3 for f(x) = 2x –x2 , in 0 <<3.

 

Here 2ℓ = 3. 

\ ℓ = 3 / 2.




 


 

Exercises

1.Obtain the Fourier series for f(x) = px in 0 < x < 2. 2.Find the Fourier series to represent x2 in the interval (-l, l ). 3.Find a Fourier series in (-2, 2), if

f(x) = 0, -2 < x < 0

 

= 1, 0 < x < 2.                                  

4.Obtain the Fourier series for         

f(x) =          1-x in 0 < x < l               

= 0 in  l      < x < 2 l.     Hence deduce that

1- (1/3 ) +(1/5) –(1/7) p/4+&  …   =

2                2        ) + (1/5        2                 2        

(1/1 ) + (1/3 )   +  p…/8) =   (

5.If f(x) = px,                          0 < x < 1              

p (2-x), 1 < x < 2,                

Show that in the interval (0,2),


6.Obtain the Fourier series for                            

f(x) = x in 0 < x < 1                                             

= 0 in 1 < x < 2                                                   

7.Obtain the Fourier series for                            

f(x) =          (cx /l ) in 0 < x < l                             

=       (c/l  ) (2 l  - x ) in l          < x < 2 l .                       

 

8.Obtain the Fourier series for

 

f(x) = (l + x ), - l < x < 0. = (l - x ), 0 < x < l.


10.Express f(x) = x  as a half –range sine series in 0 < x < 2

 

11.Obtain the half-range  sine series for ex in 0 < x < 1.

12.Find the half –range cosine series for the function f(x) = (x-2)2 in the interval 0 < x < 2.


 

 

7 Harmonic Analysis

The process of finding the Fourier series for a function given by numerical values is known as harmonic analysis.


ie, f(x) = (a0/2) + (a1 cosx + b1 sinx) + (a2 cos2x + b2 sin2x) + (a3cos3x + b3sin3x)+-------------…(1)


In (1), the term (a1cosx + b1 sinx) is called the fundamental or first harmonic, the term (a2cos2x + b2sin2x) is called the second harmonic and so on.

 

 

Example 16

Compute the first three harmonics of the Fourier series of f(x) given by the following table.


We    exclude   the   last   point   x   =   2π.

Let  f(x) = (a0/2) + (a1 cosx + b1 sinx) + (a2 cos2x + b2 sin2x) + …………

To evaluate the coefficients, we form the following table.


\ f(x) = 1.45 – 0.37cosx + 0.17 sinx – 0.1cos2x – 0.06 sin2x + 0.033 cos3x+…


 

 

 

Example 17

 

Obtain the first three coefficients in the Fourier cosine series for y, where y is given in the following table:


 

\ Fourier cosine series in the interval (0, 2π) is y = (a0 /2) + a1cosq+ a2cos2q+ a3cos3q+ …..

To evaluate the coefficients, we form the following table.


Now,         a0 = 2 (42/6) = 14

a1 = 2 ( -8.5/6) = - 2.8

a2 = 2        (-4.5/6) =

a3 = 2 (8/6) = 2.7

y = 7 –2.8 cosq- 1.5 cos2q+ 2.7 cos3q+   …..

 

Example 18

The values of x and the corresponding values of f(x) over a period T are given below. Show that  f(x) = 0.75 + 0.37 cosq+ 1.004 sinq,where q= (2πx   )/T


We omit the last value since f(x) at x = 0 is known. 

Here q= 2πx / T

When x varies from 0 to T, qvaries from   0   to   2π   with2π/6.   an   incre

 

Let  f(x) = F(q) = (a0/2) + a1 cosq+ b1 sinq.

 

To evaluate the coefficients, we form the following table.


Now, a0  = 2 ( ∑   f(x)/6)=1.5

 

a1 = 2 (1.12 /6) = 0.37

 

a2 = 2 (3.013/6) =  1.004

 

Therefore, f(x) = 0.75 + 0.37 cosq+ 1.004 sinq

 

 

Exercises

 

1.The following table gives the variations of periodic current over a period.

t (seconds)   :        0        T/6    T/3    T/2    2T/3  5T/6  T

A (amplitude): 1.98       1.30   1.05   1.30   -0.88 -0.25 1.98

Show that there is a direct current part of 0.75 amp in the variable current and obtain the

amplitude of the first harmonic.                                    

2.The turning moment  T is given for a series of values of the crank angle =75°

      :         0        30      60      90      120    150    180

T°      :         0        5224  8097  7850  5499  2626  0

Obtain the first four terms in a series of sines to represent T and calculate

T for q= 75°

3. Obtain the constant term and the co-efficient of the first sine and cosine terms in the

Fourier   expansionfollowingof table„y‟.   as   given   in   th

X       :         0        1        2        3        4        5        

Y       :         9        18      24      28      26      20      

 

4. Find the first three harmonics of Fourier series of y = f(x) from the following data. 

X : 0 ° 30° 60° 90° 120° 150° 180° 210° 240° 270° 300° 330°

 

Y : 298  356373  337     254    15580         516093       147    221    

2.8 Complex Form of Fourier Series                                       

The series for f(x) defined in the        interval (c, c+2π)and satisfying

 

 

8 Complex Form of Fourier Series


The series for f(x) defined in the        interval (c, c+2π)and satisfying

Dirichlet‟s conditions can be given in the form of 


In the interval (c, c+2ℓ), the complex  form of Fourier series is given by


 

Example 19

 

Find the complex form of the Fourier series  f(x)  = e –x  in  -1   ≤   x         ≤   1.


 

 

Example 20

Find the complex  form of the Fourier series f(x)  = ex in  - π   <   x<   π.



 

Exercises

 

Find the complex form of the Fourier series of the following functions.

 

1.f(x) = eax, -l < x < l.

 

2.f(x) = cosax, -p< x < p.

 

3.f(x) = sinx, 0 < x < p.

 

4.f(x) = e-x, -1 < x < 1.

 

5.f(x) = sinax, a is not an integer in (-pp).

 

 

 

 

 

9 SUMMARY(FOURIER SERIES)

 

 

A Fourier series of a periodic function consists of a sum of sine and cosine terms. Sines and cosines are the most fundamental periodic functions.The Fourier series is named after the French Mathematician and Physicist Jacques Fourier (1768 –1830). Fourier series has its application in problems pertaining to Heat conduction, acoustics, etc. The subject matter may be divided into the following sub topics.


 

 

FORMULA FOR FOURIER SERIES

 

Consider a real-valued function f(x) which obeys the following conditions called

Dirichlet‟s conditions   :

 

1.     f(x) is defined in an interval (a,a+2l), and f(x+2l) = f(x) so that f(x) is a periodic function of period 2l

 

2.     f(x) is continuous or has only a finite number of discontinuities in the interval (a,a+2l). 

 

3.     f(x) has no or only a finite number of maxima or minima in the interval (a,a+2l).

 

Also, let

 

is called the Fourier series of f(x) in the interval (a,a+2l). Also, the real numbers a0, a1, a2, ….an, and b1, b2 , ….bare called the Fourier coefficients of f(x). The formulae (1), (2) and   (3)   are   called   Euler‟s   formulae.

 

It can be proved that the sum of the series (4) is f(x) if f(x) is continuous at x. Thus we have


 

Suppose f(x) is discontinuous at x, then the sum of the series (4) would


 

where f(x+) and f(x-) are the values of f(x) immediately to the right and to the left of f(x) respectively.

 

Particular Cases Case (i)

 

Suppose a=0. Then f(x) is defined over the interval (0,2l). Formulae (1), (2), (3) reduce to


 

Then the right-hand side of (5) is the Fourier expansion of f(x) over the interval (0,2l).

 

If we set l=p, then f(x) is defined over the interval (0,2p). Formulae (6) reduce to


 

Also, in this case, (5) becomes


 

Case (ii)

 

Suppose a=-l. Then f(x) is defined over the interval (-l , l). Formulae (1), (2) (3) reduce to


 

Then the right-hand side of (5) is the Fourier expansion of f(x) over the interval (-l , l).

 

If we set l = p, then f(x) is defined over the interval (-pp).  Formulae (9) reduce to

 

 

 

Some useful results :

1.     The   following   rule   called   Bernoulli‟seful in evaluating the Fourier coefficients.


 

2.       The following integrals are also useful : 


 

3.                                         If   „n‟   is   integer,   then 

sin np= 0 ,       cosnp= (-1)n ,        sin2np= 0,  cos2np=1

 

 

ASSIGNMENT

 

1. The displacement y of a part of a mechanism is tabulated with corresponding angular movement x0 of the crank. Express y as a Fourier series upto the third harmonic.

 

x0      0        30      60      90      120    150    180    210    240    270    300    330

Y       1.80   1.10   0.30   0.16   1.50   1.30   2.16   1.25   1.30   1.52   1.76   2.00

 

2. Obtain the Fourier series of y upto the second harmonic using the following table :

 

 

x0      45      90      135    180    225    270    315    360

y        4.0     3.8     2.4     2.0     -1.5   0        2.8     3.4

 

3. Obtain the constant term and the coefficients of the first sine and cosine terms in the Fourier expansion of y as given in the following table :

 

x        0        1        2        3        4        5

y        9        18      24      28      26      20

 

4. Find the Fourier series of  y upto the second harmonic from the following table :

 

x        0        2        4        6        8        10      12

Y       9.0     18.2   24.4   27.8   27.5   22.0   9.0

 

5. Obtain the first 3 coefficients in the Fourier cosine series for y, where y is given below

 

x        0        1        2        3        4        5

y        4        8        15      7        6        2

 

 


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