Problem 1 - Advanced Engineering Mathematics

Question

In Problems, find the Fourier series of \(f\) on the given interval. $$ f(x)=\left\\{\begin{array}{lr} 0, & -\pi

Step-by-Step Solution

Answer

The Fourier series of the function is: \( f(x) = \frac{1}{4} + \sum_{n=1, \text{ odd}}^{\infty} \frac{2}{n\pi} \sin(nx) \).

Step 1: Determine the Function Periodicity

The function is given in the interval \(-\pi < x < \pi\). Therefore, the period of the function is \(2\pi\).

Step 2: Calculate the Coefficient a_0

The coefficient \(a_0\) is calculated using the formula \[ a_0 = \frac{1}{T} \int_{-rac{T}{2}}^{rac{T}{2}} f(x) \, dx \]For this function,\[ a_0 = \frac{1}{\pi} \left( \int_{-\pi}^{0} 0 \, dx + \int_{0}^{\pi} 1 \, dx \right) = \frac{1}{\pi} \left( 0 + \pi \right) = \frac{1}{2} \]

Step 3: Calculate the Coefficient a_n

The coefficients \(a_n\) is given by \[ a_n = \frac{2}{T} \int_{-rac{T}{2}}^{rac{T}{2}} f(x) \cos(nx) \, dx \]Since the function is 0 in the interval \(-\pi, 0\), \[ a_n = \frac{1}{\pi} \int_{0}^{\pi} \cos(nx) \, dx \]This integral evaluates to zero as it is the integral of cosine function over its period yields zero. Thus, \(a_n = 0\) for all \(n\).

Step 4: Calculate the Coefficient b_n

The coefficients \(b_n\) is given by \[ b_n = \frac{2}{T} \int_{-rac{T}{2}}^{rac{T}{2}} f(x) \sin(nx) \, dx \]For this function,\[ b_n = \frac{1}{\pi} \int_{0}^{\pi} \sin(nx) \, dx \]The integral evaluates as \[ b_n = \frac{1}{\pi} \left[-\frac{1}{n} \cos(nx) \right]_{0}^{\pi} = \frac{1}{\pi} \left(-\frac{1}{n}(-1)^n + \frac{1}{n}\right) = \begin{cases} 0 & \text{if } n \text{ is even}, \ \frac{2}{n\pi} & \text{if } n \text{ is odd} \end{cases} \]

Step 5: Write the Fourier Series

The Fourier series expansion of the function is given by:\[ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx)) \]Substitute the calculated terms,\[ f(x) = \frac{1}{4} + \sum_{n=1, \text{ odd}}^{\infty} \frac{2}{n\pi} \sin(nx) \]This represents the Fourier series of the given function.

Key Concepts

Harmonic Analysis

Harmonic analysis is a fascinating part of mathematics. It revolves around representing functions as the sum of basic wave-like functions, known as harmonics. This is done using a technique called Fourier series, which allows us to break down complex periodic functions into simpler sinusoidal components.
By doing so, we can study and analyze the behavior of the function more easily.

• What is a Harmonic? Harmonics are sinusoidal components (like sine and cosine functions) that make up a more complex signal.
• Role in Fourier Series: Harmonics help in understanding how different frequencies contribute to the shape of a function.

Each harmonic represents a fundamental frequency or its multiples, and this breakdown helps to reveal patterns and structures hidden in the data.
Harmonic analysis is widely used in engineering, physics, signal processing, and many other fields where understanding frequency patterns is essential.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals over their domain. A common example is the sine function, which repeats its values every \(2\pi\).
When dealing with Fourier series, periodic functions are essential because they can be expressed as sums of sine and cosine functions which are also periodic.
Here are some key points:

• Definition: A function \(f(x)\) is periodic if there exists a positive number \(T\) such that \(f(x + T) = f(x)\) for all \(x\).
• Period: The shortest interval \(T\) for which the function repeats is called its period. For the given function, the period is \(2\pi\).
• Why Periodicity Matters: In Fourier analysis, periodicity ensures that the function can be decomposed into a series of harmonics.

Understanding periodicity helps in predicting the behavior of functions over time. This is crucial in fields like electronics, where wave signals need to be analyzed.

Fourier Coefficients

Fourier coefficients are the numbers that make up the Fourier series representation of a function. They give us the amplitude and phase information for each harmonic component.
By calculating these coefficients, we can construct the entire Fourier series.To understand Fourier coefficients better, let's look at their types and roles:

• a_0 Coefficient: Represents the average value of the function over one period. This is the DC component in signal processing vocabulary. In our problem, \(a_0 = \frac{1}{2}\).
• a_n Coefficients: These coefficients express how much of the cosine wave of that frequency is in the function. Here, \(a_n = 0\) indicates that cosines don't contribute in this case.
• b_n Coefficients: In our example, these coefficients represent the contribution of sine components. The b_n values are found to be non-zero for odd n, hinting that only odd harmonics are present.

Each type of coefficient has a specific role that aids in deconstructing the original function. Knowing how to compute and interpret them can transform complex functions into more manageable and insightful series.