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Fourier Series: Assignment | Mathematical Methods - Physics PDF Download

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

Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics

Thus the trigonometric Fourier series of the given function is

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

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

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

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

Thus
Fourier Series: Assignment | Mathematical Methods - Physics

Hence
Fourier Series: Assignment | Mathematical Methods - Physics

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

Given Function
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics

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

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


Q.3.
Fourier Series: Assignment | Mathematical Methods - 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 | Mathematical Methods - PhysicsHence
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
= 0 for even n
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Thus, bn = 2/n for odd n
= 0 for even n
Thus the Fourier series of the given function is
Fourier Series: Assignment | Mathematical Methods - Physics

Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics


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

Fourier Series: Assignment | Mathematical Methods - Physics

Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Thus Fourier series is

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


Q.5. Find the Fourier series of
Fourier Series: Assignment | Mathematical Methods - Physics

Fourier Series: Assignment | Mathematical Methods - PhysicsFourier Series: Assignment | Mathematical Methods - Physics

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

Fourier Series: Assignment | Mathematical Methods - Physics for odd n.
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics

Now, Fourier Series: Assignment | Mathematical Methods - 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 | Mathematical Methods - Physics

Thus bn = 1/nπ , for odd n.

Fourier Series: Assignment | Mathematical Methods - Physics
Thus the Fourier series of the given function is
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics


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

The given function is
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics

Here, 2l = 2 ⇒ l = 1
Fourier Series: Assignment | Mathematical Methods - Physics

Fourier Series: Assignment | Mathematical Methods - Physics

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

Putting x = 0 gives
Fourier Series: Assignment | Mathematical Methods - 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 | Mathematical Methods - PhysicsFourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics

Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics

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

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

Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics

Fourier Series: Assignment | Mathematical Methods - Physics

For all n

Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics
Thus the Fourier Series of the given function is given by
Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics

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

 

Q.9. Find the Fourier series of 

Fourier Series: Assignment | Mathematical Methods - Physics

The given function is 

Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - PhysicsFourier Series: Assignment | Mathematical Methods - Physics
We know that
Fourier Series: Assignment | Mathematical Methods - Physics

Fourier Series: Assignment | Mathematical Methods - Physics
Fourier Series: Assignment | Mathematical Methods - Physics

We see that this formula holds for n ≠ 1.

Hence we directly calculate
Fourier Series: Assignment | Mathematical Methods - Physics

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

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

Fourier Series: Assignment | Mathematical Methods - Physics
Now we have
Fourier Series: Assignment | Mathematical Methods - Physics

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

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

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

f(x) = ex ,  0 < x < 2

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

The document Fourier Series: Assignment | Mathematical Methods - Physics is a part of the Physics Course Mathematical Methods.
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FAQs on Fourier Series: Assignment - Mathematical Methods - Physics

1. What is a Fourier series?
Ans. A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. It decomposes a periodic function into a series of harmonically related sinusoids, each with its own amplitude and phase. This series helps in analyzing and understanding the behavior of periodic functions in various applications such as signal processing, image compression, and physics.
2. How is a Fourier series calculated?
Ans. To calculate a Fourier series, the periodic function is first expressed as an infinite sum of sine and cosine functions with different frequencies. This is done by finding the coefficients of each term in the series using integral calculus. The coefficients are determined by integrating the product of the periodic function and the sine or cosine function over a period. Once the coefficients are obtained, the Fourier series can be written as a sum of these terms, which represents the original periodic function.
3. What are the applications of Fourier series?
Ans. Fourier series has a wide range of applications in various fields. Some of the major applications include: - Signal Processing: Fourier series is extensively used in analyzing and manipulating signals in fields like telecommunications, audio processing, and image processing. - Physics: It is utilized in the study of wave phenomena, such as analyzing the vibrations of strings, sound waves, and electromagnetic waves. - Image Compression: Fourier series plays a crucial role in image compression algorithms like JPEG, where it helps in reducing the amount of data required to represent an image without significant loss of quality. - Quantum Mechanics: It is used to analyze and understand the behavior of quantum mechanical systems, particularly in solving the Schrödinger equation for various potentials.
4. Can any periodic function be represented by a Fourier series?
Ans. Yes, any periodic function with a finite number of discontinuities and bounded variation can be represented by a Fourier series. However, the convergence of the Fourier series depends on the smoothness of the function and the presence of any abrupt changes or sharp corners. Functions with discontinuities or sharp corners may exhibit a phenomenon called Gibbs phenomenon, where the Fourier series approximation has oscillations near the discontinuity. In such cases, additional terms or modifications may be needed to accurately represent the function.
5. Are there any limitations or drawbacks of using Fourier series?
Ans. While Fourier series is a powerful tool for analyzing periodic functions, it does have some limitations and drawbacks. - Convergence: The convergence of a Fourier series may be an issue for functions with discontinuities or sharp corners. The Gibbs phenomenon can lead to oscillations near the discontinuity, affecting the accuracy of the representation. - Non-periodic Functions: Fourier series is specifically designed for periodic functions and may not be suitable for the analysis of non-periodic functions. Other techniques like Fourier transforms or wavelet transforms are better suited for non-periodic functions. - Finite Precision: In practical applications, the accuracy of the Fourier series representation may be limited by the finite precision of numerical calculations. Round-off errors and truncation errors can affect the fidelity of the representation. - Complexity: Calculating the coefficients and manipulating the Fourier series can be computationally intensive, especially for functions with complex or irregular shapes. This can pose challenges in real-time applications or when dealing with large datasets.
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