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Half Range Expansion of a Fourier series:-
Even and odd functions:-
Even Function:-
A function f(x) is called even if f(-x) = f(x) for all value of x.
Properties of even functions:-
The graph of an even function is always symmetrized about the y-axis.
f(x) contains only even power of x and may contains only cosx,sec x.
Sum of two even function is even.
Product of two even function is even.
𝑎
−𝑎
𝑎
0
Odd Function:-
A function f(x) is called odd if f(-x) = - f(x) ,∀ x∊ℝ
The graph of an odd function is always symmetric about the origin
and lie in opposite quadrant (
st
and 3
rd
f(x) contains only odd power of x and may contains only sin x, cosec
x.
𝑎
−𝑎
Sum of two odd function is odd function.
Product of an odd function and even function is an odd function.
Product of two odd function is an even function.
Neither Even nor Odd functions:-
A function f(x) which is not satisfying the even and odd condition is called
neither even nor odd function.
Some Useful Formulas:-
Sin nπ= cos (n+1/2) π=0 ∀ n €ℤ
i.e. sin π = sin 2π =sin (3π =sin (-π)=sin(-2π)=……………..=
i.e. cos π/2=cos 3π/2=cos 5π/2=…………………………………..=
cos nπ =sin(n+1/2) π=(− 1 )
𝑛
∀ n €ℤ
Cos π= cos 3π =cos 5π=……………………………………= - 1
Sin π/2=sin 5π/2 =sin 9π/2=…=
Cos 0=cos 2π=cos 4π=---=
Sin 3π/2=sin 7π/2=sin 11π/2=…- 1
If x=𝑥
0
is the part of the finite discontinuity then sum of the fourier
series
f(x)=
1
2
[lim
ℎ→ 0
0
− ℎ) + lim
ℎ→ 0
0
+ ℎ)]
f(x)=1/
[
0
0
)]
Even and odd Extensions of a Function:-
Let f be a function given on the interval (0,a),we define the even/odd
extension of f to be even/odd functions on the interval (-a,a),which
coincides with f on the half interval (0,a).
Even Extension:-
𝑬
Odd Extension:-
𝟎
Half Range Expansion of a Fourier series:-
Suppose a function is defined in the range(0,L), instead of the full range (-
L,L).Then the expansion f(x) contains in a series of sine or cosine terms only
.The series is termed as half range sine series or half range cosine series.
If f(x) is taken to be an odd function, its Fourier series expansion will
consists of only sine terms. Hence the Fourier series expansion of f(x)
represents “Half range expansion of Fourier sine series”.
Ans:-Given that f(x)=sin x is defined in (0 ,𝜋 ) ie L=π
0
𝑛
cos
∞
𝑛= 1
0
𝑛
cos nx
∞
𝑛= 1
Now
0
∫ sin 𝑥 𝑑𝑥 =
[− cos 𝑥]
0
𝜋
𝜋
0
𝜋
0
𝑛
cos 𝑛𝑥 𝑑𝑥 =
𝜋
0
∫ sin 𝑥 cos 𝑛𝑥 𝑑𝑥
𝜋
0
2
𝜋(𝑛
2
− 1 )
[
𝑛
]
𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 n≠
By definition 𝑎
𝑛
4
𝜋(𝑛
2
− 1 )
Hence, the half range cosine series is given by
f
x
π
π
cosnx
n
2
∞
n= 2
n
i. e sin 𝑥 =
2
π
4
π
(cos
2 𝑥
1 ∗ 3
4 𝑥
3 ∗ 5
6 𝑥
5 ∗ 7
required cosine series.
Exercise 2.
Are the following functions odd, even, neither odd nor even
(a) 𝑥
2
cos 𝑥(b)sin x +cos x (c)f(x)=𝑒
𝑥
2
(d) ln x (e) 𝑒
|𝑥|
(f) f(x)=𝑥
4
(0<x<2π)
(g)𝑓
cos
2
sin
2
(h) 𝑒
−|𝑥|
(-π<x<π)
- Determine half range Fourier sine series for f(x)=x(π-x) in 0<x< π
hence deduce
1
1
3
1
3
3
1
5
3
3
2
- Determine half range Fourier sine series for f(x)=e^x in 0<x<π
4.Determine the half range Fourier sine series for
f(x)={
𝜋
2
𝜋
2
- Determine the half range Fourier cosine series for
f(x)= {
𝜋
2
𝜋
2
6.Find half-range Fourier cosine series for f(x)=x in (0<x<2π)
