Fourier Series Mathematics Notes | EduRev

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Mathematics : Fourier Series Mathematics Notes | EduRev

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FOURIER SERIES 
Fourier series, piecewise continuous functions, Fourier cosine and sine series, property of Fourier 
coefficient, Fourier theorem, discussion of the theorem and its corollary. 
 Reference:  
 John P. D’ Angelo, An introduction to Complex Analysis and Geometry, American Mathematical Society, 2010. 
Fourier Series, Lecture notes published by the Institute of life Long Learning, University of Delhi, Delhi, 2011. 
Fourier Series  
A particle is said to be periodic function with period ? , if f(x± ?) =f(x) ? x 
Result  
1. ? f(x)dx
 
 
 = ? f(x)dx
    
    
 
2. ? f(x)dx
 
  
= ? f(x)dx
    
 
 
3. ? f(x)dx
 
  
=? f(  x)dx
 
  
 
4. ? cosmxcosnx dx
 
  
 ={
0
 
2 
for m n
        for m n 0
        for m n 0
 
5. ? sinmxsinnx 
 
  
= {
0
 
0
for m n
       for m n 0
       for m n 0
 
6. ? cosmxsinnx
 
  
=0 for all m, n +ve integer 
7 ? cosnx dx
 
  
 = 2
0
2  
         
if n 0 
if n 0
 
8 ? sinnx dx
 
  
 = 0 
When the French mathematician Joseph Fourier (1768-1830) was trying to solve a problem in heat 
conduction, he needed to express a function f as an infinite series of sine and cosine function (called a 
trigonometric series) 
f(x) a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
  .(i) 
We start by assuming [assume – to think or accept that sth is true but without having proof of it] that the 
trigonometric series converges uniformly on ,  , - to the continuous sum function f(x): 
we integrate both sides of equation (i),using term by term integration valid due 
to uniform convergence,we obtain 
? f(x)dx
 
  
 ? a
 
dx
 
  
 ? ?(a
 
cosnx b
 
sinnx)
 
   
 
  
dx 
 a
 
.2  ? a
 
 
   
? cosnx
 
  
dx ? b
 
? sinnxdx
 
  
 
   
 
 a
 
.2  0 0 
a
 
 
1
2 
? f(x)dx
 
  
      .(ii) 
To determine a n  for n 1 ,we multiply both sides of equation  
f(x) a
 
 ?(a
 
cosmx b
 
sinmx)
 
   
  .(iii) 
bycosnx (since,|cosnx| 1 ,therefore uniform convergence of the series is not  
destroyed,and so term by term of integration is justified) and intergrate term 
by term from   to    
justified (in doing sth) having a good reason for doing sth 
? f(x)cosnxdx
 
  
 a
 
? cosnxdx
 
  
 ? a
 
cosmxcosnxdx
 
   
 ? b
 
? sinmx cosnxdx
 
  
 
   
 
 0 a
 
  0 
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FOURIER SERIES 
Fourier series, piecewise continuous functions, Fourier cosine and sine series, property of Fourier 
coefficient, Fourier theorem, discussion of the theorem and its corollary. 
 Reference:  
 John P. D’ Angelo, An introduction to Complex Analysis and Geometry, American Mathematical Society, 2010. 
Fourier Series, Lecture notes published by the Institute of life Long Learning, University of Delhi, Delhi, 2011. 
Fourier Series  
A particle is said to be periodic function with period ? , if f(x± ?) =f(x) ? x 
Result  
1. ? f(x)dx
 
 
 = ? f(x)dx
    
    
 
2. ? f(x)dx
 
  
= ? f(x)dx
    
 
 
3. ? f(x)dx
 
  
=? f(  x)dx
 
  
 
4. ? cosmxcosnx dx
 
  
 ={
0
 
2 
for m n
        for m n 0
        for m n 0
 
5. ? sinmxsinnx 
 
  
= {
0
 
0
for m n
       for m n 0
       for m n 0
 
6. ? cosmxsinnx
 
  
=0 for all m, n +ve integer 
7 ? cosnx dx
 
  
 = 2
0
2  
         
if n 0 
if n 0
 
8 ? sinnx dx
 
  
 = 0 
When the French mathematician Joseph Fourier (1768-1830) was trying to solve a problem in heat 
conduction, he needed to express a function f as an infinite series of sine and cosine function (called a 
trigonometric series) 
f(x) a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
  .(i) 
We start by assuming [assume – to think or accept that sth is true but without having proof of it] that the 
trigonometric series converges uniformly on ,  , - to the continuous sum function f(x): 
we integrate both sides of equation (i),using term by term integration valid due 
to uniform convergence,we obtain 
? f(x)dx
 
  
 ? a
 
dx
 
  
 ? ?(a
 
cosnx b
 
sinnx)
 
   
 
  
dx 
 a
 
.2  ? a
 
 
   
? cosnx
 
  
dx ? b
 
? sinnxdx
 
  
 
   
 
 a
 
.2  0 0 
a
 
 
1
2 
? f(x)dx
 
  
      .(ii) 
To determine a n  for n 1 ,we multiply both sides of equation  
f(x) a
 
 ?(a
 
cosmx b
 
sinmx)
 
   
  .(iii) 
bycosnx (since,|cosnx| 1 ,therefore uniform convergence of the series is not  
destroyed,and so term by term of integration is justified) and intergrate term 
by term from   to    
justified (in doing sth) having a good reason for doing sth 
? f(x)cosnxdx
 
  
 a
 
? cosnxdx
 
  
 ? a
 
cosmxcosnxdx
 
   
 ? b
 
? sinmx cosnxdx
 
  
 
   
 
 0 a
 
  0 
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 a
 
 
1
 
? f(x)cosnxdx,
 
  
n 1 ,2,3, . eq
 
(iv) 
similary,if we multiply both sides of the eq
 
 (iii) bysinnx and integrate from 
   to  ,we get 
b
 
 
1
 
? f(x)sinnx
 
  
dx,n 1 ,2,3,  eq
 
(v) 
Definition A trigonometric series of the form 
a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
  ..(vi) 
Is called Fouries series of a function f periodic with 2  if the Fourier coefficients a 0, a n, b n are determined by 
(ii) , (iii) & (v) 
note Since,|a
 
cosnx b
 
sinnx| |a
 
| |b
 
|,  n 1 
by  ei erstrass
 
s,M test,it follows that,if the series ?(|a
 
| |b
 
|)
 
   
converges, 
then the trigonometric series (vi)converges absolutely and uniformly in ,  , - 
and it is the Fourier series of a continuous 2  periodic fuction f(x).Moreover 
?(a
 
 
 b
 
 
)   and so lim
   
a
 
 0 , lim
    
b
 
 0 . 
Alternative way of defining Fourier Series 
A series is 
 
 
 
 +? (a
 
 
   
cosnx b
 
sinnx) is called a trigonometric series and the constant a 0, a n &b n are 
called coefficient. If this series converges its sum is periodic and its period is 2 .  
(ii) let f  be bounded and integrable on ,  ,   - then the series  
 
 
 
 +? (a
 
 
   
cosnx b
 
sinnx) where  
            a n = 
 
 
 ? f(x)cosnxdx
 
  
  ?  n  0, 1, 2 .. 
and b n = 
 
 
? f(x)sinnxdx
 
  
 ?  n  0, 1, 2 ..   (1) 
is called the Fourier series of the function on ,  ,    ] and the coefficient is given by the equation (1) are 
called Fourier coefficient of the function. 
Imp.thm. Let  f   R  R be such that  
(i) f  is bounded and integrable on ,  ,   ] 
(ii) The function f  is piece wise monotonic on ,  ,   ] 
(iii) f  is periodic with period 2  i.e.  f(x+2 ) = f(x)  ? x ? R  
then a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
 
{
 
 
 
 
f(x),for every point x of continuity in ,  , -
1
2
,f(x
 
) f(x
 
)-,for   x  
1
2
,f( 
 
) f(  
 
)-,for x ±       
 
Note that the trigonometric series (and hence called then Fourier series) converges to the average of the left-
hand and right-hand limits of the function f at points x in -  , , of discontinuity, of the function f at points x 
in -  , , of discontinuity,and 
using periodicity of f,i.e.f(x 2 ) f(x),
1
2
,f( 
 
) f(  
 
)-at the end points ±  
Theorem: 
Let f : R   R be such that  
(i) f  is bounded and integrable on ,  ,   ] 
(ii) f  is monotonic on each of ( ?? , 0) and (0, ?? ) where (0 < ?? <  ) then 
 
 
 
 +? a
 
 
   
= 
 ( 
 
)  ( 
 
)
 
 
M. Imp. Que: Expand in a series of sine and cosine of multiply of x, the formula given by 
 f(x) = 2
x  
  x
              
   x 0
0 x  
  
what is the sum of the fourier series function x  0 and x      . Deduce that  
 
 
 
 = 1+
 
 
 
 
 
 
 
 
 
 
 
           (1) 
Check SV notes pg 11  
Sol
n
: f(x) is bounded and integrable on ,  ,    ]    
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FOURIER SERIES 
Fourier series, piecewise continuous functions, Fourier cosine and sine series, property of Fourier 
coefficient, Fourier theorem, discussion of the theorem and its corollary. 
 Reference:  
 John P. D’ Angelo, An introduction to Complex Analysis and Geometry, American Mathematical Society, 2010. 
Fourier Series, Lecture notes published by the Institute of life Long Learning, University of Delhi, Delhi, 2011. 
Fourier Series  
A particle is said to be periodic function with period ? , if f(x± ?) =f(x) ? x 
Result  
1. ? f(x)dx
 
 
 = ? f(x)dx
    
    
 
2. ? f(x)dx
 
  
= ? f(x)dx
    
 
 
3. ? f(x)dx
 
  
=? f(  x)dx
 
  
 
4. ? cosmxcosnx dx
 
  
 ={
0
 
2 
for m n
        for m n 0
        for m n 0
 
5. ? sinmxsinnx 
 
  
= {
0
 
0
for m n
       for m n 0
       for m n 0
 
6. ? cosmxsinnx
 
  
=0 for all m, n +ve integer 
7 ? cosnx dx
 
  
 = 2
0
2  
         
if n 0 
if n 0
 
8 ? sinnx dx
 
  
 = 0 
When the French mathematician Joseph Fourier (1768-1830) was trying to solve a problem in heat 
conduction, he needed to express a function f as an infinite series of sine and cosine function (called a 
trigonometric series) 
f(x) a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
  .(i) 
We start by assuming [assume – to think or accept that sth is true but without having proof of it] that the 
trigonometric series converges uniformly on ,  , - to the continuous sum function f(x): 
we integrate both sides of equation (i),using term by term integration valid due 
to uniform convergence,we obtain 
? f(x)dx
 
  
 ? a
 
dx
 
  
 ? ?(a
 
cosnx b
 
sinnx)
 
   
 
  
dx 
 a
 
.2  ? a
 
 
   
? cosnx
 
  
dx ? b
 
? sinnxdx
 
  
 
   
 
 a
 
.2  0 0 
a
 
 
1
2 
? f(x)dx
 
  
      .(ii) 
To determine a n  for n 1 ,we multiply both sides of equation  
f(x) a
 
 ?(a
 
cosmx b
 
sinmx)
 
   
  .(iii) 
bycosnx (since,|cosnx| 1 ,therefore uniform convergence of the series is not  
destroyed,and so term by term of integration is justified) and intergrate term 
by term from   to    
justified (in doing sth) having a good reason for doing sth 
? f(x)cosnxdx
 
  
 a
 
? cosnxdx
 
  
 ? a
 
cosmxcosnxdx
 
   
 ? b
 
? sinmx cosnxdx
 
  
 
   
 
 0 a
 
  0 
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 a
 
 
1
 
? f(x)cosnxdx,
 
  
n 1 ,2,3, . eq
 
(iv) 
similary,if we multiply both sides of the eq
 
 (iii) bysinnx and integrate from 
   to  ,we get 
b
 
 
1
 
? f(x)sinnx
 
  
dx,n 1 ,2,3,  eq
 
(v) 
Definition A trigonometric series of the form 
a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
  ..(vi) 
Is called Fouries series of a function f periodic with 2  if the Fourier coefficients a 0, a n, b n are determined by 
(ii) , (iii) & (v) 
note Since,|a
 
cosnx b
 
sinnx| |a
 
| |b
 
|,  n 1 
by  ei erstrass
 
s,M test,it follows that,if the series ?(|a
 
| |b
 
|)
 
   
converges, 
then the trigonometric series (vi)converges absolutely and uniformly in ,  , - 
and it is the Fourier series of a continuous 2  periodic fuction f(x).Moreover 
?(a
 
 
 b
 
 
)   and so lim
   
a
 
 0 , lim
    
b
 
 0 . 
Alternative way of defining Fourier Series 
A series is 
 
 
 
 +? (a
 
 
   
cosnx b
 
sinnx) is called a trigonometric series and the constant a 0, a n &b n are 
called coefficient. If this series converges its sum is periodic and its period is 2 .  
(ii) let f  be bounded and integrable on ,  ,   - then the series  
 
 
 
 +? (a
 
 
   
cosnx b
 
sinnx) where  
            a n = 
 
 
 ? f(x)cosnxdx
 
  
  ?  n  0, 1, 2 .. 
and b n = 
 
 
? f(x)sinnxdx
 
  
 ?  n  0, 1, 2 ..   (1) 
is called the Fourier series of the function on ,  ,    ] and the coefficient is given by the equation (1) are 
called Fourier coefficient of the function. 
Imp.thm. Let  f   R  R be such that  
(i) f  is bounded and integrable on ,  ,   ] 
(ii) The function f  is piece wise monotonic on ,  ,   ] 
(iii) f  is periodic with period 2  i.e.  f(x+2 ) = f(x)  ? x ? R  
then a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
 
{
 
 
 
 
f(x),for every point x of continuity in ,  , -
1
2
,f(x
 
) f(x
 
)-,for   x  
1
2
,f( 
 
) f(  
 
)-,for x ±       
 
Note that the trigonometric series (and hence called then Fourier series) converges to the average of the left-
hand and right-hand limits of the function f at points x in -  , , of discontinuity, of the function f at points x 
in -  , , of discontinuity,and 
using periodicity of f,i.e.f(x 2 ) f(x),
1
2
,f( 
 
) f(  
 
)-at the end points ±  
Theorem: 
Let f : R   R be such that  
(i) f  is bounded and integrable on ,  ,   ] 
(ii) f  is monotonic on each of ( ?? , 0) and (0, ?? ) where (0 < ?? <  ) then 
 
 
 
 +? a
 
 
   
= 
 ( 
 
)  ( 
 
)
 
 
M. Imp. Que: Expand in a series of sine and cosine of multiply of x, the formula given by 
 f(x) = 2
x  
  x
              
   x 0
0 x  
  
what is the sum of the fourier series function x  0 and x      . Deduce that  
 
 
 
 = 1+
 
 
 
 
 
 
 
 
 
 
 
           (1) 
Check SV notes pg 11  
Sol
n
: f(x) is bounded and integrable on ,  ,    ]    
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  a
 
  
 
 
 ? f(x)cosnxdx
 
  
  
        =  
 
 
0? f(x)cosnxdx ? f(x)cosnxdx
 
 
 
  
1 
 =  
 
 
0? (x  )cosnxdx ? (  x)cosnxdx
 
 
 
  
1=
 
 
I 1+I 2   (i)  
Consider,         I 1 =
 
 
?
(x  )cos nx dx
 
  
 
           Let         u      x      ?  dx    du 
                           I 1   =?
( u  )cosn( u) ( du)
 
 
 
 
 = ? (u  )cos nu du
 
 
   ?
(u  )cos nu du
 
 
   ?
(  x) cos nx dx
 
 
 
                    a n =
 
 
?
(   x   x) cosnx dx
 
 
    (ii) 
   
 
 
? xcosnx dx
 
 
 
   
 
 
0.x
      
 
/
 
 
 ? 1 .
     
 
dx
 
 
1 
   
 
 
0
 
 
 
(cosnx)
 
 
1   
 
  
 
,cosn  cos0 -  
 
  
 
,( 1 )
 
 1 - 
                       a n = 
 
  
 
,1 ( 1 )
 
- 
Put n = 0 in equation (ii) 
  a
 
 =  
 
 
? x dx
 
 
   
 
 
.
 
 
(x
 
)
 
 
  
Similarly,  
 b n = 
 
 
*1 ( 1 )
 
+ 
b n  = 
 
 
? f(x)sinnx dx
 
  
 
 =
 
 
? f(x)sinnx dx
 
  
 +? f(x)sin nx dx
 
 
 
 =
 
 
? (x  )sin nx dx
 
  
 +? (  x)sinnx dx
 
 
 
Consider, I 1 = ? (x  )sinnx dx
 
  
 
Let u   x  ? du    dx 
     I 1 = ? ( u  )sinn( u)( du)
 
 
 
   ? (u  )sin nudu
 
 
 
 =? (u  )sin nudu
 
 
 
 =? (x  )sinnxdx
 
 
       by change of variable 
  b n  =
 
 
[?
(x     x)sinnxdx
 
 
]   
 = 
 
 
? 2 sinnxdx
 
 
 = 20 
     
 
1
 
 
  2 0
(  )
 
 
 
 
 
1 = 
 
 
,1 ( 1 )
 
- 
  b n =
 
 
,1 ( 1 )
 
- 
? a n = 
 
 
 
 
 
 
 ,0 ,
 
 
 
 ,0 ,
 
 
 
,0 ,     
? b n = 4 1 ,0 ,
 
 
 ,0 ,
 
 
,0 ,     
  the fourier series of the function f(x) 
f(x)  
 
 
+02
 
 
     
 
 
 
 
 
      
 
 
 
 
 
 
       
 
 
  3 2
     
 
 4
      
 
 
       
 
  31  (iii) 
The sum of the fourier series at x 0 is 
 ( 
 
)  ( 
 
)
 
  
 =
    
 
 =0 ,? f(0
 
)= lim
   
 
f(x)= lim
   
 
(x ?? )= lim
   
(0 h ?? )  ??  
The sum of the fourier series at x = ±  
  is 
 (  
 
)  ( 
 
)
 
 
lim
   
 
f(x)= lim
   
 
(?? x)= lim
   
?? (?? h ) = 0 
    0?  lim
    
 
 f(x) lim
    
 
(x  ) lim
   
(   h  )  2 1 
Sum of the fourier series at x 0 is f(0) 
   put x 0 in equation (iii) 
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FOURIER SERIES 
Fourier series, piecewise continuous functions, Fourier cosine and sine series, property of Fourier 
coefficient, Fourier theorem, discussion of the theorem and its corollary. 
 Reference:  
 John P. D’ Angelo, An introduction to Complex Analysis and Geometry, American Mathematical Society, 2010. 
Fourier Series, Lecture notes published by the Institute of life Long Learning, University of Delhi, Delhi, 2011. 
Fourier Series  
A particle is said to be periodic function with period ? , if f(x± ?) =f(x) ? x 
Result  
1. ? f(x)dx
 
 
 = ? f(x)dx
    
    
 
2. ? f(x)dx
 
  
= ? f(x)dx
    
 
 
3. ? f(x)dx
 
  
=? f(  x)dx
 
  
 
4. ? cosmxcosnx dx
 
  
 ={
0
 
2 
for m n
        for m n 0
        for m n 0
 
5. ? sinmxsinnx 
 
  
= {
0
 
0
for m n
       for m n 0
       for m n 0
 
6. ? cosmxsinnx
 
  
=0 for all m, n +ve integer 
7 ? cosnx dx
 
  
 = 2
0
2  
         
if n 0 
if n 0
 
8 ? sinnx dx
 
  
 = 0 
When the French mathematician Joseph Fourier (1768-1830) was trying to solve a problem in heat 
conduction, he needed to express a function f as an infinite series of sine and cosine function (called a 
trigonometric series) 
f(x) a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
  .(i) 
We start by assuming [assume – to think or accept that sth is true but without having proof of it] that the 
trigonometric series converges uniformly on ,  , - to the continuous sum function f(x): 
we integrate both sides of equation (i),using term by term integration valid due 
to uniform convergence,we obtain 
? f(x)dx
 
  
 ? a
 
dx
 
  
 ? ?(a
 
cosnx b
 
sinnx)
 
   
 
  
dx 
 a
 
.2  ? a
 
 
   
? cosnx
 
  
dx ? b
 
? sinnxdx
 
  
 
   
 
 a
 
.2  0 0 
a
 
 
1
2 
? f(x)dx
 
  
      .(ii) 
To determine a n  for n 1 ,we multiply both sides of equation  
f(x) a
 
 ?(a
 
cosmx b
 
sinmx)
 
   
  .(iii) 
bycosnx (since,|cosnx| 1 ,therefore uniform convergence of the series is not  
destroyed,and so term by term of integration is justified) and intergrate term 
by term from   to    
justified (in doing sth) having a good reason for doing sth 
? f(x)cosnxdx
 
  
 a
 
? cosnxdx
 
  
 ? a
 
cosmxcosnxdx
 
   
 ? b
 
? sinmx cosnxdx
 
  
 
   
 
 0 a
 
  0 
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 a
 
 
1
 
? f(x)cosnxdx,
 
  
n 1 ,2,3, . eq
 
(iv) 
similary,if we multiply both sides of the eq
 
 (iii) bysinnx and integrate from 
   to  ,we get 
b
 
 
1
 
? f(x)sinnx
 
  
dx,n 1 ,2,3,  eq
 
(v) 
Definition A trigonometric series of the form 
a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
  ..(vi) 
Is called Fouries series of a function f periodic with 2  if the Fourier coefficients a 0, a n, b n are determined by 
(ii) , (iii) & (v) 
note Since,|a
 
cosnx b
 
sinnx| |a
 
| |b
 
|,  n 1 
by  ei erstrass
 
s,M test,it follows that,if the series ?(|a
 
| |b
 
|)
 
   
converges, 
then the trigonometric series (vi)converges absolutely and uniformly in ,  , - 
and it is the Fourier series of a continuous 2  periodic fuction f(x).Moreover 
?(a
 
 
 b
 
 
)   and so lim
   
a
 
 0 , lim
    
b
 
 0 . 
Alternative way of defining Fourier Series 
A series is 
 
 
 
 +? (a
 
 
   
cosnx b
 
sinnx) is called a trigonometric series and the constant a 0, a n &b n are 
called coefficient. If this series converges its sum is periodic and its period is 2 .  
(ii) let f  be bounded and integrable on ,  ,   - then the series  
 
 
 
 +? (a
 
 
   
cosnx b
 
sinnx) where  
            a n = 
 
 
 ? f(x)cosnxdx
 
  
  ?  n  0, 1, 2 .. 
and b n = 
 
 
? f(x)sinnxdx
 
  
 ?  n  0, 1, 2 ..   (1) 
is called the Fourier series of the function on ,  ,    ] and the coefficient is given by the equation (1) are 
called Fourier coefficient of the function. 
Imp.thm. Let  f   R  R be such that  
(i) f  is bounded and integrable on ,  ,   ] 
(ii) The function f  is piece wise monotonic on ,  ,   ] 
(iii) f  is periodic with period 2  i.e.  f(x+2 ) = f(x)  ? x ? R  
then a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
 
{
 
 
 
 
f(x),for every point x of continuity in ,  , -
1
2
,f(x
 
) f(x
 
)-,for   x  
1
2
,f( 
 
) f(  
 
)-,for x ±       
 
Note that the trigonometric series (and hence called then Fourier series) converges to the average of the left-
hand and right-hand limits of the function f at points x in -  , , of discontinuity, of the function f at points x 
in -  , , of discontinuity,and 
using periodicity of f,i.e.f(x 2 ) f(x),
1
2
,f( 
 
) f(  
 
)-at the end points ±  
Theorem: 
Let f : R   R be such that  
(i) f  is bounded and integrable on ,  ,   ] 
(ii) f  is monotonic on each of ( ?? , 0) and (0, ?? ) where (0 < ?? <  ) then 
 
 
 
 +? a
 
 
   
= 
 ( 
 
)  ( 
 
)
 
 
M. Imp. Que: Expand in a series of sine and cosine of multiply of x, the formula given by 
 f(x) = 2
x  
  x
              
   x 0
0 x  
  
what is the sum of the fourier series function x  0 and x      . Deduce that  
 
 
 
 = 1+
 
 
 
 
 
 
 
 
 
 
 
           (1) 
Check SV notes pg 11  
Sol
n
: f(x) is bounded and integrable on ,  ,    ]    
Free coaching of B.Sc (h) maths & JAM 
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  a
 
  
 
 
 ? f(x)cosnxdx
 
  
  
        =  
 
 
0? f(x)cosnxdx ? f(x)cosnxdx
 
 
 
  
1 
 =  
 
 
0? (x  )cosnxdx ? (  x)cosnxdx
 
 
 
  
1=
 
 
I 1+I 2   (i)  
Consider,         I 1 =
 
 
?
(x  )cos nx dx
 
  
 
           Let         u      x      ?  dx    du 
                           I 1   =?
( u  )cosn( u) ( du)
 
 
 
 
 = ? (u  )cos nu du
 
 
   ?
(u  )cos nu du
 
 
   ?
(  x) cos nx dx
 
 
 
                    a n =
 
 
?
(   x   x) cosnx dx
 
 
    (ii) 
   
 
 
? xcosnx dx
 
 
 
   
 
 
0.x
      
 
/
 
 
 ? 1 .
     
 
dx
 
 
1 
   
 
 
0
 
 
 
(cosnx)
 
 
1   
 
  
 
,cosn  cos0 -  
 
  
 
,( 1 )
 
 1 - 
                       a n = 
 
  
 
,1 ( 1 )
 
- 
Put n = 0 in equation (ii) 
  a
 
 =  
 
 
? x dx
 
 
   
 
 
.
 
 
(x
 
)
 
 
  
Similarly,  
 b n = 
 
 
*1 ( 1 )
 
+ 
b n  = 
 
 
? f(x)sinnx dx
 
  
 
 =
 
 
? f(x)sinnx dx
 
  
 +? f(x)sin nx dx
 
 
 
 =
 
 
? (x  )sin nx dx
 
  
 +? (  x)sinnx dx
 
 
 
Consider, I 1 = ? (x  )sinnx dx
 
  
 
Let u   x  ? du    dx 
     I 1 = ? ( u  )sinn( u)( du)
 
 
 
   ? (u  )sin nudu
 
 
 
 =? (u  )sin nudu
 
 
 
 =? (x  )sinnxdx
 
 
       by change of variable 
  b n  =
 
 
[?
(x     x)sinnxdx
 
 
]   
 = 
 
 
? 2 sinnxdx
 
 
 = 20 
     
 
1
 
 
  2 0
(  )
 
 
 
 
 
1 = 
 
 
,1 ( 1 )
 
- 
  b n =
 
 
,1 ( 1 )
 
- 
? a n = 
 
 
 
 
 
 
 ,0 ,
 
 
 
 ,0 ,
 
 
 
,0 ,     
? b n = 4 1 ,0 ,
 
 
 ,0 ,
 
 
,0 ,     
  the fourier series of the function f(x) 
f(x)  
 
 
+02
 
 
     
 
 
 
 
 
      
 
 
 
 
 
 
       
 
 
  3 2
     
 
 4
      
 
 
       
 
  31  (iii) 
The sum of the fourier series at x 0 is 
 ( 
 
)  ( 
 
)
 
  
 =
    
 
 =0 ,? f(0
 
)= lim
   
 
f(x)= lim
   
 
(x ?? )= lim
   
(0 h ?? )  ??  
The sum of the fourier series at x = ±  
  is 
 (  
 
)  ( 
 
)
 
 
lim
   
 
f(x)= lim
   
 
(?? x)= lim
   
?? (?? h ) = 0 
    0?  lim
    
 
 f(x) lim
    
 
(x  ) lim
   
(   h  )  2 1 
Sum of the fourier series at x 0 is f(0) 
   put x 0 in equation (iii) 
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f(0)  =  
 
 
 
 
 
01 
 
 
 
 
 
 
 
  1+0 
0 = 
 
 
 
 
 
01 
 
 
 
 
 
 
 
  1+0 
 
 
.
 
 
 = 1 
 
 
 
 
 
 
 
   
    
 
 
 
 = 1 
 
 
 
 
 
 
 
   
Que. If f(x)=2
 cosx
cosx
               
   x 0
0 x  
 
P/T  the fourier series of the function f(x) is 
 
 
 
 0
 
 . 
sin2x  
 
 . 
sin4x 
 
 . 
sin6x   1 
Sol
n
 Since the function f(x) is bounded and integrable on , ?? , ?? ]  
      a
 
  
 
 
 ? f(x)cosnxdx
 
  
  
 =
 
 
0? f(x) cosnxdx
 
  
 
 
 
? f(x)cosnxdx
 
 
  1 
 = 
 
 
I 1+
 
 
I 2 
Consider, 
    I 1 = ? f(x) cosnxdx
 
  
 
      u    x ? dx   du 
   I 1 =? cos( u)cosn( u)( du)
 
 
 
   ? cosucosnudu
 
 
 
 =? cosucosnudu
 
 
 
 =? cosxcosnxdx
 
 
 
  a n   
 
 
? cosxcosnxdx
 
 
+
 
 
? cosxcosnxdx
 
 
 
 = 0 
b n = 
 
 
? f(x)sinnxdx
 
  
 
 = 
 
 
? cosxsinnxdx
 
  
+
 
 
? cosxsinnxdx
 
 
 
Let I 1=
 
 
? cosxsinnxdx
 
  
 
              
 
 
? cos( u)sinn( u)( du)
 
 
  Put   x   u ,  dx   du                                                                                                                                                                                                                                                                 
 =
 
 
? cosusinnu du
 
 
 
 =
 
 
? cosxsinnxdx
 
 
 
   b n =
 
 
? cosxsinnxdx
 
 
 
 =
 
 
?
,sin(n 1 )x sin(n 1 )x-dx
 
 
 
 =
 
 
0 
   (   ) 
   
 
   (   ) 
   
1
 
 
 
   
 
 
0
   (   ) 
   
 
   (   ) 
   
1
 
 
 
   
 
 
 0
 
   
*cos(n 1 )  cos0 + 
 
   
*cos(n 1 )  cos0 +1 
   
 
 
 0
 
   
*( 1 )
   
 1 + 
 
   
*( 1 )
   
 1 +1 
   
 
 
,( 1 )
   
 1 -0
 
   
 
 
   
1  *?      ( 1 )
   
 ( 1 )
   
 
   
 
 
,( 1 )
   
 1 -
  
( 
 
  )
 
 =
 
 
.
 
 
 
  
,1 ( 1 )
   
- 
 =
 
 
.
 
 
 
  
,1 ( 1 )
 
( 1 )- 
b n = 
 
 
.
 
 
 
  
,1 ( 1 )
 
- 
b n =.0 ,
 
 
,
 
 
 
  
,0 ,
 
 
,
 
 
 
  
,0 ./ 
Since, a 0=0 and a n=0 
  the fourier series is  
f(x) = ? b
 
 
   
 sin nx 
Page 5


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FOURIER SERIES 
Fourier series, piecewise continuous functions, Fourier cosine and sine series, property of Fourier 
coefficient, Fourier theorem, discussion of the theorem and its corollary. 
 Reference:  
 John P. D’ Angelo, An introduction to Complex Analysis and Geometry, American Mathematical Society, 2010. 
Fourier Series, Lecture notes published by the Institute of life Long Learning, University of Delhi, Delhi, 2011. 
Fourier Series  
A particle is said to be periodic function with period ? , if f(x± ?) =f(x) ? x 
Result  
1. ? f(x)dx
 
 
 = ? f(x)dx
    
    
 
2. ? f(x)dx
 
  
= ? f(x)dx
    
 
 
3. ? f(x)dx
 
  
=? f(  x)dx
 
  
 
4. ? cosmxcosnx dx
 
  
 ={
0
 
2 
for m n
        for m n 0
        for m n 0
 
5. ? sinmxsinnx 
 
  
= {
0
 
0
for m n
       for m n 0
       for m n 0
 
6. ? cosmxsinnx
 
  
=0 for all m, n +ve integer 
7 ? cosnx dx
 
  
 = 2
0
2  
         
if n 0 
if n 0
 
8 ? sinnx dx
 
  
 = 0 
When the French mathematician Joseph Fourier (1768-1830) was trying to solve a problem in heat 
conduction, he needed to express a function f as an infinite series of sine and cosine function (called a 
trigonometric series) 
f(x) a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
  .(i) 
We start by assuming [assume – to think or accept that sth is true but without having proof of it] that the 
trigonometric series converges uniformly on ,  , - to the continuous sum function f(x): 
we integrate both sides of equation (i),using term by term integration valid due 
to uniform convergence,we obtain 
? f(x)dx
 
  
 ? a
 
dx
 
  
 ? ?(a
 
cosnx b
 
sinnx)
 
   
 
  
dx 
 a
 
.2  ? a
 
 
   
? cosnx
 
  
dx ? b
 
? sinnxdx
 
  
 
   
 
 a
 
.2  0 0 
a
 
 
1
2 
? f(x)dx
 
  
      .(ii) 
To determine a n  for n 1 ,we multiply both sides of equation  
f(x) a
 
 ?(a
 
cosmx b
 
sinmx)
 
   
  .(iii) 
bycosnx (since,|cosnx| 1 ,therefore uniform convergence of the series is not  
destroyed,and so term by term of integration is justified) and intergrate term 
by term from   to    
justified (in doing sth) having a good reason for doing sth 
? f(x)cosnxdx
 
  
 a
 
? cosnxdx
 
  
 ? a
 
cosmxcosnxdx
 
   
 ? b
 
? sinmx cosnxdx
 
  
 
   
 
 0 a
 
  0 
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 a
 
 
1
 
? f(x)cosnxdx,
 
  
n 1 ,2,3, . eq
 
(iv) 
similary,if we multiply both sides of the eq
 
 (iii) bysinnx and integrate from 
   to  ,we get 
b
 
 
1
 
? f(x)sinnx
 
  
dx,n 1 ,2,3,  eq
 
(v) 
Definition A trigonometric series of the form 
a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
  ..(vi) 
Is called Fouries series of a function f periodic with 2  if the Fourier coefficients a 0, a n, b n are determined by 
(ii) , (iii) & (v) 
note Since,|a
 
cosnx b
 
sinnx| |a
 
| |b
 
|,  n 1 
by  ei erstrass
 
s,M test,it follows that,if the series ?(|a
 
| |b
 
|)
 
   
converges, 
then the trigonometric series (vi)converges absolutely and uniformly in ,  , - 
and it is the Fourier series of a continuous 2  periodic fuction f(x).Moreover 
?(a
 
 
 b
 
 
)   and so lim
   
a
 
 0 , lim
    
b
 
 0 . 
Alternative way of defining Fourier Series 
A series is 
 
 
 
 +? (a
 
 
   
cosnx b
 
sinnx) is called a trigonometric series and the constant a 0, a n &b n are 
called coefficient. If this series converges its sum is periodic and its period is 2 .  
(ii) let f  be bounded and integrable on ,  ,   - then the series  
 
 
 
 +? (a
 
 
   
cosnx b
 
sinnx) where  
            a n = 
 
 
 ? f(x)cosnxdx
 
  
  ?  n  0, 1, 2 .. 
and b n = 
 
 
? f(x)sinnxdx
 
  
 ?  n  0, 1, 2 ..   (1) 
is called the Fourier series of the function on ,  ,    ] and the coefficient is given by the equation (1) are 
called Fourier coefficient of the function. 
Imp.thm. Let  f   R  R be such that  
(i) f  is bounded and integrable on ,  ,   ] 
(ii) The function f  is piece wise monotonic on ,  ,   ] 
(iii) f  is periodic with period 2  i.e.  f(x+2 ) = f(x)  ? x ? R  
then a
 
 ?(a
 
cosnx b
 
sinnx)
 
   
 
{
 
 
 
 
f(x),for every point x of continuity in ,  , -
1
2
,f(x
 
) f(x
 
)-,for   x  
1
2
,f( 
 
) f(  
 
)-,for x ±       
 
Note that the trigonometric series (and hence called then Fourier series) converges to the average of the left-
hand and right-hand limits of the function f at points x in -  , , of discontinuity, of the function f at points x 
in -  , , of discontinuity,and 
using periodicity of f,i.e.f(x 2 ) f(x),
1
2
,f( 
 
) f(  
 
)-at the end points ±  
Theorem: 
Let f : R   R be such that  
(i) f  is bounded and integrable on ,  ,   ] 
(ii) f  is monotonic on each of ( ?? , 0) and (0, ?? ) where (0 < ?? <  ) then 
 
 
 
 +? a
 
 
   
= 
 ( 
 
)  ( 
 
)
 
 
M. Imp. Que: Expand in a series of sine and cosine of multiply of x, the formula given by 
 f(x) = 2
x  
  x
              
   x 0
0 x  
  
what is the sum of the fourier series function x  0 and x      . Deduce that  
 
 
 
 = 1+
 
 
 
 
 
 
 
 
 
 
 
           (1) 
Check SV notes pg 11  
Sol
n
: f(x) is bounded and integrable on ,  ,    ]    
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  a
 
  
 
 
 ? f(x)cosnxdx
 
  
  
        =  
 
 
0? f(x)cosnxdx ? f(x)cosnxdx
 
 
 
  
1 
 =  
 
 
0? (x  )cosnxdx ? (  x)cosnxdx
 
 
 
  
1=
 
 
I 1+I 2   (i)  
Consider,         I 1 =
 
 
?
(x  )cos nx dx
 
  
 
           Let         u      x      ?  dx    du 
                           I 1   =?
( u  )cosn( u) ( du)
 
 
 
 
 = ? (u  )cos nu du
 
 
   ?
(u  )cos nu du
 
 
   ?
(  x) cos nx dx
 
 
 
                    a n =
 
 
?
(   x   x) cosnx dx
 
 
    (ii) 
   
 
 
? xcosnx dx
 
 
 
   
 
 
0.x
      
 
/
 
 
 ? 1 .
     
 
dx
 
 
1 
   
 
 
0
 
 
 
(cosnx)
 
 
1   
 
  
 
,cosn  cos0 -  
 
  
 
,( 1 )
 
 1 - 
                       a n = 
 
  
 
,1 ( 1 )
 
- 
Put n = 0 in equation (ii) 
  a
 
 =  
 
 
? x dx
 
 
   
 
 
.
 
 
(x
 
)
 
 
  
Similarly,  
 b n = 
 
 
*1 ( 1 )
 
+ 
b n  = 
 
 
? f(x)sinnx dx
 
  
 
 =
 
 
? f(x)sinnx dx
 
  
 +? f(x)sin nx dx
 
 
 
 =
 
 
? (x  )sin nx dx
 
  
 +? (  x)sinnx dx
 
 
 
Consider, I 1 = ? (x  )sinnx dx
 
  
 
Let u   x  ? du    dx 
     I 1 = ? ( u  )sinn( u)( du)
 
 
 
   ? (u  )sin nudu
 
 
 
 =? (u  )sin nudu
 
 
 
 =? (x  )sinnxdx
 
 
       by change of variable 
  b n  =
 
 
[?
(x     x)sinnxdx
 
 
]   
 = 
 
 
? 2 sinnxdx
 
 
 = 20 
     
 
1
 
 
  2 0
(  )
 
 
 
 
 
1 = 
 
 
,1 ( 1 )
 
- 
  b n =
 
 
,1 ( 1 )
 
- 
? a n = 
 
 
 
 
 
 
 ,0 ,
 
 
 
 ,0 ,
 
 
 
,0 ,     
? b n = 4 1 ,0 ,
 
 
 ,0 ,
 
 
,0 ,     
  the fourier series of the function f(x) 
f(x)  
 
 
+02
 
 
     
 
 
 
 
 
      
 
 
 
 
 
 
       
 
 
  3 2
     
 
 4
      
 
 
       
 
  31  (iii) 
The sum of the fourier series at x 0 is 
 ( 
 
)  ( 
 
)
 
  
 =
    
 
 =0 ,? f(0
 
)= lim
   
 
f(x)= lim
   
 
(x ?? )= lim
   
(0 h ?? )  ??  
The sum of the fourier series at x = ±  
  is 
 (  
 
)  ( 
 
)
 
 
lim
   
 
f(x)= lim
   
 
(?? x)= lim
   
?? (?? h ) = 0 
    0?  lim
    
 
 f(x) lim
    
 
(x  ) lim
   
(   h  )  2 1 
Sum of the fourier series at x 0 is f(0) 
   put x 0 in equation (iii) 
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f(0)  =  
 
 
 
 
 
01 
 
 
 
 
 
 
 
  1+0 
0 = 
 
 
 
 
 
01 
 
 
 
 
 
 
 
  1+0 
 
 
.
 
 
 = 1 
 
 
 
 
 
 
 
   
    
 
 
 
 = 1 
 
 
 
 
 
 
 
   
Que. If f(x)=2
 cosx
cosx
               
   x 0
0 x  
 
P/T  the fourier series of the function f(x) is 
 
 
 
 0
 
 . 
sin2x  
 
 . 
sin4x 
 
 . 
sin6x   1 
Sol
n
 Since the function f(x) is bounded and integrable on , ?? , ?? ]  
      a
 
  
 
 
 ? f(x)cosnxdx
 
  
  
 =
 
 
0? f(x) cosnxdx
 
  
 
 
 
? f(x)cosnxdx
 
 
  1 
 = 
 
 
I 1+
 
 
I 2 
Consider, 
    I 1 = ? f(x) cosnxdx
 
  
 
      u    x ? dx   du 
   I 1 =? cos( u)cosn( u)( du)
 
 
 
   ? cosucosnudu
 
 
 
 =? cosucosnudu
 
 
 
 =? cosxcosnxdx
 
 
 
  a n   
 
 
? cosxcosnxdx
 
 
+
 
 
? cosxcosnxdx
 
 
 
 = 0 
b n = 
 
 
? f(x)sinnxdx
 
  
 
 = 
 
 
? cosxsinnxdx
 
  
+
 
 
? cosxsinnxdx
 
 
 
Let I 1=
 
 
? cosxsinnxdx
 
  
 
              
 
 
? cos( u)sinn( u)( du)
 
 
  Put   x   u ,  dx   du                                                                                                                                                                                                                                                                 
 =
 
 
? cosusinnu du
 
 
 
 =
 
 
? cosxsinnxdx
 
 
 
   b n =
 
 
? cosxsinnxdx
 
 
 
 =
 
 
?
,sin(n 1 )x sin(n 1 )x-dx
 
 
 
 =
 
 
0 
   (   ) 
   
 
   (   ) 
   
1
 
 
 
   
 
 
0
   (   ) 
   
 
   (   ) 
   
1
 
 
 
   
 
 
 0
 
   
*cos(n 1 )  cos0 + 
 
   
*cos(n 1 )  cos0 +1 
   
 
 
 0
 
   
*( 1 )
   
 1 + 
 
   
*( 1 )
   
 1 +1 
   
 
 
,( 1 )
   
 1 -0
 
   
 
 
   
1  *?      ( 1 )
   
 ( 1 )
   
 
   
 
 
,( 1 )
   
 1 -
  
( 
 
  )
 
 =
 
 
.
 
 
 
  
,1 ( 1 )
   
- 
 =
 
 
.
 
 
 
  
,1 ( 1 )
 
( 1 )- 
b n = 
 
 
.
 
 
 
  
,1 ( 1 )
 
- 
b n =.0 ,
 
 
,
 
 
 
  
,0 ,
 
 
,
 
 
 
  
,0 ./ 
Since, a 0=0 and a n=0 
  the fourier series is  
f(x) = ? b
 
 
   
 sin nx 
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 =
 
 
0
 
 . 
sin2x 
 
 . 
sin4x  1 
Que. Show that  
 Cos kx =
      
 
 0
 
 
 
      
 
 
  
 
 
       
 
 
  
 
  1    ??   x   ??  
k being non integr al. Deduce that  
? ?? cot kx = 
 
 
 2k ?
 
 
 
  
 
 
   
 
? 
 
     
  ? ( 1 )
 
0
 
   
 
 
     
1
 
   
 
Sol
n
 f(x)= cos kx   ??   x  ??  
   a n =  
 
 
 ? f(x)cosnxdx
 
  
  
 =
 
 
 ? coskxcosnxdx
 
  
 
 =
 
 
 ? coskxcosnxdx
 
 
 
 =
 
 
? cos(k n) cos(k n)xdx
 
 
 
Now,   sin(k+n)?? = sin k?? cos n?? +cos k?? sin n??  
 =( 1 )
 
sin k?? + 0 
 =( 1 )
 
sin k?? 
Similary, sin(k n)?? = ( 1 )
 
 sin k??  
  a n =
(  )
 
     
 
0
 
   
 
 
   
1 
 =
(  )
 
     
 
0
  
(   )(   )
1 
  a 0 =
     
 
.
  
 
 
 = 
     
 
.
 
 
 
b n =
 
 
? f(x)sinnx dx
 
  
 = 
 
 
? coskxsinnx dx 
 
  
   0    *? of odd function + 
   b n = 0 
The fourier series of the function f(x) is  
a
 
2
 ? a
 
cosnx
 
   
 
sink 
 
.
1
k
 ?
( 1 )
 
sink 
 
 
   
.
2k
k
 
 n
 
cosnx 
 =
      
  
+
       
 
 ?
(  )
 
 
 
  
 
cosnx
 
   
 
Cos kx=
     
 
0
 
 
 2k ?
(  )
 
 
 
  
 
 
   
cosnx 1 
Since , the function f(x) is continuous on , ?? , ?? ] 
   fourier series at x  f(x) ? x? , ?? , ?? ]. 
Deduction  
(i)In particular the fourier series at ?? =f(?? ) 
 Cos k?? =
     
 
0
 
 
 2k ?
(  )
 
 
 
  
 
cosn 
 
   
1 
?? cot k?? =
 
 
 2k ?
(  )
 
(  )
 
 
 
  
 
 
   
 
?? cot k?? =
 
 
 2k ?
 
 
 
  
 
 
   
 
(ii)In particular the fourier series at 0=f(0) 
Cos 0 = 
     
 
0
 
 
 2k ?
(  )
 
 
 
  
 
 
   
1 
 
     
  
 
 
 ?
(  )
 
  
 
 
  
 
 
   
 
 =
 
 
 ? ( 1 )
 
0
 
   
 
 
   
1
 
   
 
 =2
 
 
 ? ( 1 )
 
.
 
   
 
   
3 ?
(  )
 
 
   
 
    
 
 =2? ( 1 )
 
 
   
 
    
3 ? ( 1 )
 
 
(     )
 
   
 
 =? ( 1 )
 
0
 
   
 
 
(     )
1
 
   
 
Que let  
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