Topic: Fourier Series ( Periodic Function to change of interval)

AEM (2130002)

ALA Project

Topic: Fourier Series ( Periodic Function to change of interval)

Guided By: Prof. Bhoomika Ma’am

BRANCH: ELECTRICAL DIV: B

Prepared By:Himal Desai 140120109008

Abhishek Choksi 140120109005

Soham Davara 140120109007

Fourier series• Periodic function that occur in many physical and

engineering problems for example, in conduction of heat and mechanical vibration are complicated and it is desirable to represent them in a series of sine and cosines. Most of the single valued function can be expressed in a trigonometric series of the form

+ ---------- (1)

• Within a desired range of value of the variable such as a series is known as Fourier Series

• Thus the function f(x) defined in the interval c ≤ x ≤ c + 2 can be expressed in the Fourier Series

f(x) = -------- (2)• Where , ( n = 1,2,3,…) are constant, called the

Fourier coefficients of f(x) are required to be determined.

Change Of Interval• In many engineering problems, it is required to

expand a function in a Fourier series over an interval of length 2l instead of 2 .

• The transformation from the function of period p = 2 to those of period p = 2l is quite simple.

• This can be achieved by transformation of the variable.

• Consider a periodic function f(x) defined in the interval c ≤ x ≤ c + 2l.

• To change the interval into length 2 .

• Put z = So that when x = c, z = = d and when x = c + 2l, z = = + 2 = d + 2 • Thus the function f(x) of period 2l in c to c + 2l is

transformed to the function.• f() = f(z) of the period 2 in d to d + 2 and f(z) can

be expressed as the Fourier series.

PROOF:• F(z) = ---------- (1)• Where, , n= 1,2,3… , n= 1,2,3…Now making the inverse substitution z = , dz = dxWhen, z = d, x = cand when, z = d + 2 , x = c + 2l

• The expression 1 becomesf(z) = f() = f(x) = • Thus the Fourier series for f(x) in the interval c to

c + 2l is given by, f(x) = ------ (2)Where, , n= 1,2,3… , n= 1,2,3…

Example: Obtain Fourier series for the function f(x) = x, 0 ≤ x ≤ 1 p = 2l = 2 = (2 - x) 1 ≤ x ≤ 2• Solution: Let f(x) = Where,

= = + = =

= = + • Since sin n = sin 2n = 0, cos 2n = 1 for all n =

1,2,3…. = = = 0 if n is even = if n is odd

= = = = 0 ANS: f(x) =

Example: Find the Fourier series with period 3 to represent f(x) = , in the range (0,3)

• Solution:Here p = 2l = 3So, l = For this period 2l = 3, we have

f(x) = Where,

= =

=

• Since sin 2n = sin 0 = 0 cos 2n = cos 0 = 1 for all n = 1,2,3….

= =

= =

 ANS:f(x) =

Example : Find the corresponding Fourier seriesF(x) = p = 2l = 4

• Solution:Here 2l = 4So, l = 2Let,

f(x) = Where,

= = =

= = = 0

=

= = = = = , if n is odd = 0 , if n is even

ANS:f(x) =

THANK YOU