Unit  2
Fourier Series
In last semester we find expansions of functions by using Taylor’s series as well as Maclaurin’s series. Now here we express a special type of function in terms of sines and cosines are called as a Fourier Series.
Periodic Function:
A function is called periodic if it is defined every real x and if there exist a some positive integer ‘T’ such that , n = 1, 2, 3, ……
The number ‘T’ is called period of the function f(x). i.e. The graph of f(x) repeats after the interval T.
e.g. sin x or cos x are periodic functions of period .
Note:
If ‘T’ is a period of function f(x) then 2T, 3T, 4T, ….. are also periods of f(x). The smallest of all these is called as primitive period of f(x).
Important formulas to evaluating integrals
Where dashes indicates derivatives and suffixes indicates integrations. 3. 2. sin A. cos B = sin (A + B) + sin (A – B) 2. cos A. sin B = sin (A + B) + sin (A – B) 2. cos A. cos B = cos (A + B) + cos (A – B) 2. sin A. sin B = cos (A – B) – cos (A + B) 4. 5. 6.
7. 8. 9. and Takes the values 1, 0, 1, 0, 1, 0, …. Depending on the values of n.

Even and odd functions
 Even functions:
A function f(x) is defined on the both sided interval i.e. on (l, l) or or or is said to be even if.
Also if
; then f(x) is called odd function.
e.g.
(i) f(x) = x2 is even function
(ii) f(x) = sin x is odd function.
Note:
1) The addition or subtraction of two odd functions is again odd.
2) The addition or subtraction of two even functions is even.
3) The addition or subtraction of even and odd function neither odd nor even.
4) The product of two even functions is again even.
5) The product of two odd functions is also even.
6) The product of even and odd functions is odd.
Let f(x) be a function defined in C < x < C + 2L such that
1) f(x) is defined and single valued in the given internal also. exist.
2) f(x) may have finite number of finite discontinuities in the interval.
3) f(x) may have finite number of maxima or minima in the given interval.
Definition:
Fourier Series
Let f(x) be a periodic function of period 2L. defined in the internal and satisfied Dirichlet's conditions, then f(x) can be expressed as,
.
Where ao, an, bn are called Fourier constant’s or Fourier coefficients and are given by,
Note:
That there are only 4 intervals as below. i.e. is divided into following four intervals.
Note that for [0, 2L] we put c = 0 
Hence Fourier series in this interval will be, 
Where 
Simillarly, for the interval we put c = 0, 
Hence Fourier series in this interval will be 
Where 
Note that for the interval [L, L] i.e. put C = L, 
First we check whether f(x) is even function or odd. 
Case I:
If f(x) is even function. Then we get half range cosine series as,
Where
Case II:
If f(x) is odd function. Then we get half range sine series as,
Where
Simillarly
Note that for that interval i.e. put ,
First we check wheatear f(x) is even or odd function.
Case I:
If f(x) is even function then we get half range cosine series as
Where
Case II:
If f(x) is odd function then we get half range sine series as,
Where
Note that
 For half range cosine series i.e. f(x) is even function bn = 0
 For half range sine series i.e. f(x) is odd function ao = an = 0
Q1)
Find the Fourier series of f(x) = x in the interval
S1)
Here ; 
It’s Fourier series is given by 
… (1) 
Where 

& 
Hence the required Fourier series is 
Q1) Find the Fourier series for
in the interval
Hence deduce that
S1)
Here ; 
Hence it’s Fourier series is, 
… (1) 
Where 
& 
Hence equation (1) becomes 
Put we get 
i.e. 

Q2) Find a Fourier series expansion in the interval for
;
;
S2)
Here 
; 
; 
Hence it’s Fourier series expansion is, 
… (1) 
Where 
And 
Hence equation (1) becomes 

Q3) Find a Fourier series of
;
;
S3)
Here 
; 
; 
Here f(x) is odd function Hence we get half range sine series i.e. 
… (1) 
Where 
Hence equation (1) becomes, 
Q4) Find a Fourier series for
;
S4)
Here 
; 
Since f(x) is even function hence 
It’s Fourier series is 
… (1) 
Where 
Hence equation (1) becomes, 
Q5) Find half range cosine series of in the interval and hence deduce that
a)
b)
S5)
Here 
; 
Hence it’s half range cosine series is, 
… (1) 
Where 
Hence equation (1) becomes, 
… (2) 
Put x = 0, we get 
Hence the result 
Put we get, 
i.e. 

The Fourier expansion of any function f(x) in the interval is given as, 
… (1) 
Where 
& 
Practically these functions are often not given by formula but by table of corresponding values. In such cases integral is cannot be evaluated and Hence we use mean value of the function over the range is 
Thus the above formulas becomes, 
Where m is the no of subintervals 
Simillarly, 
Here the term in (1) is 
is called the fundamental or first harmonic and the term is called second harmonic and so on. 
Note that
 The amplitude of nth harmonic is
 The square of amplitude is called energy of the nth harmonic.
i.e. is called energy of the nth harmonic
3. % of nth harmonic
Exercise
 Find Fourier series for
in & Hence deduce that
2. Find a Fourier series for
in the internal & hence deduce that
3. Find a Fourier series for
,
Hence, show that
a)
b)
c)
Find the Fourier series of the Function
;
Find Half range cosine series of and Hence deduce that
a)
b)
Find half range sine series of in .
Reference Books:
1. Advanced Engineering Mathematics by Erwin Kreyszig (Wiley Eastern Ltd.)
2. Advanced Engineering Mathematics by M. D. Greenberg (Pearson Education)
3. Advanced Engineering Mathematics by Peter V. O’Neil (Thomson Learning)
4. Thomas’ Calculus by George B. Thomas, (AddisonWesley, Pearson)
5. Applied Mathematics (Vol. I & Vol. II) by P.N.Wartikar and J.N.Wartikar Vidyarthi Griha Prakashan, Pune.
6. Linear Algebra –An Introduction, Ron Larson, David C. Falvo (Cenage Learning, Indian edition)