This document provides information about Fourier series and even/odd functions from the Jahangirbad Institute of Technology. It defines Fourier series and even/odd functions, lists their properties, and provides 14 important questions about finding Fourier series for various periodic functions over different intervals. The questions aim to find the Fourier series and deduce trigonometric identities.

1. JAHANGIRABAD INSTITUTE
OF
TECHNOLOGY
FOURER SERIES
(QUESTION BANK)
ENGINEERING MATHEMATICS –II
PREPRAED BY
MOHAMMAD IMRAN
(ASSISTANT PROFESSOR, JIT)
E-mail: mohammad.imran@jit.edu.in
Website: www.jit.edu.in
Mobile no 9648588546

2. FOURER SERIES
DEFINITION :
An infinite Series of Trigonometric functions which represents an expansion or approximation of a
periodic function, used in Fourier analysis.
EVEN FUNCTION:
Let f(x) be a real-valued function of a real variable.
Then f is evenif the following equation holds for all
x and -x in the domain of f.
( ) ( )f x f x 
or
( ) ( ) 0f x f x  
Geometrically speaking, the graph face of an even function
is symmetric with respect to the y-axis, meaning that its
graph remains unchanged after reflection
about the y-axis.
Examples of even functions are |x|, x2, x4, cos(x), cosh(x),
or any linear combination of these.
ODD FUNCTIONS
Again, let f(x) be a real-valued function of a real variable.
Then f is odd if the following equation holds for all
x and -x in the domain of f:
( ) ( )f x f x  
or
( ) ( ) 0f x f x  
Geometrically, the graph of an odd function has rotational
symmetry with respect to the origin, meaning that its graph
remains unchanged after rotation of 180 degrees about the
origin.

3. Examples of odd functions are x, x3, sin(x), sinh(x), erf(x), or
any linear combination of these.
Properties involving addition and subtraction
 The sum of two even functions is even, and any constant multiple of an even function is even.
 The sum of two odd functions is odd, and any constant multiple of an odd function is odd.
 The difference between two odd functions is odd.
 The difference between two even functions is even.
 The sum of an even and odd function is neither even nor odd, unless one of the functions is equal to zero over
the given domain.
Properties involving multiplication and division
 The product of two even functions is an even function.
 The product of two odd functions is an even function.
 The product of an even function and an odd function is an odd function.
 The quotient of two even functions is an even function.
 The quotient of two odd functions is an even function.
 The quotient of an even function and an odd function is an odd function.
Properties involving composition
 The composition of two even functions is even.
 The composition of two odd functions is odd.
 The composition of an even function and an odd function is even.
 The composition of any function with an even function is even (but not vice versa).
Other algebraic properties
 Any linear combination of even functions is even, and the even functions form a vector space over the reals.
Similarly, any linear combination of odd functions is odd, and the odd functions also form a vector space over
the reals. In fact, the vector space of all real-valued functions is the direct sum of the subspaces of even and
odd functions. In other words, every function f(x) can be written uniquely as the sum of an even function and
an odd function:
( ) ( ) ( )e of x f x f x 

4. where
 
1
( ) ( ) ( )
2
ef x f x f x  
is even and
 
1
( ) ( ) ( )
2
of x f x f x  
is odd. For example, if f is exp, then fe is cosh and fo is sinh.
 The even functions form a commutative algebra over the reals. However, the odd functions do not form an
algebra over the reals, as they are not closed under multiplication.

17. IMPORTANT QUESTIONS
1. find the Fourier series for the function
2
( )
4
x
f x x x      uptu 2009
2. find the Fourier series for the function ( ) sin 0 2f x x x x    uptu 2001
3. find the Fourier series for the function
0
( )
0
k x
f x
k x


   
 
 
uptu 2010
hence show that
1 1 1
1 ........
3 5 7 4

    
4. find the Fourier series for the function
0
( )
0
x x
f x
x x


  
 
  
uptu 2002, 2008
Hence show that
2
2 2 2 2
1 1 1 1
........
1 3 5 7 8

    
5. find the Fourier series for the function
2
( )
4
x
f x x x      uptu 2009
6. find the Fourier series for the function 2
( )f x x x x      uptu 2003
Deduce that
2
2 2 2 2
1 1 1 1
........
6 1 2 3 4

    
7. find the Fourier series for the function
0
( ) ( 2 ) ( )
0
x x
f x and f x f x
x x
 

 
   
  
   
uptu 2006
8. find the Fourier series for the function
1 0
( ) ( 2 ) ( )
1 0
x
f x and f x f x
x



   
  
 
9. find the Fourier series for the function ( ) sinf x x x x     uptu 2008
Deduce that
1 1 1 1 2
........
1.3 3.5 5.7 7.9 4
 
    
10. find the Fourier series for the function ( ) cosf x x x x     uptu 2008
11. Obtain the half range sine series for the function 2
( )f x x in the interval 0 3x  uptu 2008
12. Find the half range cosine series for the function
2 0 1
( )
2(2 ) 1 2
t x
f x
t x
 
 
  
uptu 2001,06, 07
13. Obtain the fouriercosine seriesexpansionofthe periodicfunctiondefine bythe function uptu 2001
( ) sin , 0 1
t
f t t
l
 
   
 
14. Find the half range sine series for the function ( )f x given in the range (0,1) by the graph OPQ as shown
in the figure. Uptu 2009