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.
If f(x) defined in the interval (c, c+2π ), then the Parseval‟s Identity is given by
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
Find the Fourier series expansion for the function
Find the Fourier series of periodicity 3 for f(x) = 2x –x2 , in 0 <x <3.
Here 2ℓ = 3.
\ ℓ = 3 / 2.
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.