Fourier Series: Assignment

Q.1. Expand  in trigonometric Fourier series. Draw three periods and use it to derive the following relations.

Thus the trigonometric Fourier series of the given function is

The graph of the function is shown in the figure

(i) At x = 0 , the Fourier series converges to

(ii) At x = π we have

(iii) we have at x = π/2

Thus

Hence

Q.2. Expand the given function in Fourier Series  and hence find the value of

Given Function

Here, 2l = 2 ⇒ l = 1

Thus, bn = 4/nπ for odd n.

= 0 for even n.
Hence the Fourier Series of the given function is

Q.3.
The 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
Hence

= 0 for even n

Thus, bn = 2/n for odd n
= 0 for even n
Thus the Fourier series of the given function is

Q.4. Expand the given function in Fourier Series  and hence find the value of

Thus Fourier series is

At x = 1, f (x) = 2

Q.5. Find the Fourier series of

Now,  for n = 1, 5, 9, 13
for 3, 7,11,15
and  for even n.
Thus, an = 1/nπ  for n = 1, 5, 9, 13
for n 3, 7,11,15
We can write the general formula for  for even n and

for odd n.

Now,  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

Thus bn = 1/nπ , for odd n.

Thus the Fourier series of the given function is

Q.6. Expand the given function in Fourier Series  and hence find the value of

The given function is

Here, 2l = 2 ⇒ l = 1

for odd n and an a = for even n.

Thus the Fourier Series of the given function is

Putting x = 0 gives

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

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

Thus, bn = -4π/n for all n.
Thus the Fourier series of the given function is

Q.8. Expand the given function in Fourier Series f(x) = π sinπx, 0 < x < 1 and hence find the value of

Here, 2l = 1 ⇒ l = 1/2

For all n

Thus the Fourier Series of the given function is given by

At x = 0 , the function is discontinuous and the fourier series converges to

Q.9. Find the Fourier series of

The given function is

We know that

We see that this formula holds for n ≠ 1.

Hence we directly calculate

Now let us evaluate an using equation (I)

when n is odd, (n + 1) is even and (n - 1) is even. Thus for odd n,

When n is even (n + 1) and (n - 1) are odd, hence

Thus an = 0 for odd n

Now we have

We see that we cannot calculate b1 using this formula.
Let us evaluate b1 directly

Let us evaluate other bn ’s using equation (ii)

Thus bn = 0 for all n.

Hence the Fourier Series of the given function is

In the expanded form

Q.10. Expand the given functions in Fourier Series f(x) = ex ,  0 < x < 2 and hence find the value of

f(x) = ex ,  0 < x < 2

Hence,

Thus the Fourier Series of the given function is

In the expanded form we can write

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

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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