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Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

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Q.1. Expand Fourier Series: Assignment - Notes | Study Mathematical Models - Physics in trigonometric Fourier series. Draw three periods and use it to derive the following relations.
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Thus the trigonometric Fourier series of the given function is

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
The graph of the function is shown in the figure
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

(i) At x = 0 , the Fourier series converges to
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
(ii) At x = π we have

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
(iii) we have at x = π/2
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Thus
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Hence
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Q.2. Expand the given function in Fourier Series Fourier Series: Assignment - Notes | Study Mathematical Models - Physics and hence find the value of Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Given Function
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Here, 2l = 2 ⇒ l = 1
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Thus, bn = 4/nπ for odd n.

= 0 for even n.
Hence the Fourier Series of the given function is
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics


Q.3.
Fourier Series: Assignment - Notes | Study Mathematical Models - PhysicsThe function shown in the graph is plotted for one period. Sketch it over three periods and use the graph to find the Fourier series of the function shown.

f (x) = -(x + π), -π < x < 0 

x, 0 < x < π
Fourier Series: Assignment - Notes | Study Mathematical Models - PhysicsHence
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
= 0 for even n
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Thus, bn = 2/n for odd n
= 0 for even n
Thus the Fourier series of the given function is
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics


Q.4. Expand the given function in Fourier Series Fourier Series: Assignment - Notes | Study Mathematical Models - Physics and hence find the value of Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Thus Fourier series is

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
At x = 1, f (x) = 2
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics


Q.5. Find the Fourier series of
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - PhysicsFourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Now, Fourier Series: Assignment - Notes | Study Mathematical Models - Physics for n = 1, 5, 9, 13
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics for 3, 7,11,15
and Fourier Series: Assignment - Notes | Study Mathematical Models - Physics for even n.
Thus, an = 1/nπ  for n = 1, 5, 9, 13
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics for n 3, 7,11,15
We can write the general formula for Fourier Series: Assignment - Notes | Study Mathematical Models - Physics for even n and 

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics for odd n.
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Now, Fourier Series: Assignment - Notes | Study Mathematical Models - Physics for 2, 6,10,14....
= +1 for n = 4, 8,12,16 ......
= 0 for odd n
Hence bn = 2/nπ  for  n = 2, 6,10,14....
= 0 for n = 4, 8, 12, 16 ....

= 1/nπ for odd n.
We can write the general formula for even n
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Thus bn = 1/nπ , for odd n.

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Thus the Fourier series of the given function is
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics


Q.6. Expand the given function in Fourier Series Fourier Series: Assignment - Notes | Study Mathematical Models - Physics and hence find the value of Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

The given function is
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Here, 2l = 2 ⇒ l = 1
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics for odd n and an a = for even n.
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Thus the Fourier Series of the given function is
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Putting x = 0 gives
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics


Q.7. Find the Fourier series of f (x) = x2 , 0 < x < 2π

The given function is f (x) = x2 , 0 < x < 2π

Fourier Series: Assignment - Notes | Study Mathematical Models - PhysicsFourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Thus, bn = -4π/n for all n.
Thus the Fourier series of the given function is
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics


Q.8. Expand the given function in Fourier Series f(x) = π sinπx, 0 < x < 1 and hence find the value of Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - PhysicsHere, 2l = 1 ⇒ l = 1/2
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

For all n

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Thus the Fourier Series of the given function is given by
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

At x = 0 , the function is discontinuous and the fourier series converges to
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics 

 

Q.9. Find the Fourier series of 

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

The given function is 

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - PhysicsFourier Series: Assignment - Notes | Study Mathematical Models - Physics
We know that
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

We see that this formula holds for n ≠ 1.

Hence we directly calculate
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Now let us evaluate an using equation (I)
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
when n is odd, (n + 1) is even and (n - 1) is even. Thus for odd n,

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
When n is even (n + 1) and (n - 1) are odd, hence
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Thus an = 0 for odd n
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Now we have
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
We see that we cannot calculate b1 using this formula.
Let us evaluate b1 directly
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Let us evaluate other bn ’s using equation (ii)
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Thus bn = 0 for all n.

Hence the Fourier Series of the given function is
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
In the expanded form
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics


Q.10. Expand the given functions in Fourier Series f(x) = ex ,  0 < x < 2 and hence find the value of Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

f(x) = ex ,  0 < x < 2

Fourier Series: Assignment - Notes | Study Mathematical Models - PhysicsFourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Hence,
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
Thus the Fourier Series of the given function is
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics
In the expanded form we can write
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

At x = 0, the function is discontinuous and the Fourier series converges to the average of its left hand and right hand limit.
Thus , at x = 0
Fourier Series: Assignment - Notes | Study Mathematical Models - Physics

The document Fourier Series: Assignment - Notes | Study Mathematical Models - Physics is a part of the Physics Course Mathematical Models.
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