Problem 13 - Advanced Engineering Mathematics

Question

Since \(f(x)\) is an even function, we expand in a cosine series: $$\begin{aligned} &a_{0}=\frac{2}{\pi} \int_{0}^{\pi} x d x=\pi\\\ &a_{n}=\frac{2}{\pi} \int_{0}^{\pi} x \cos n x d x=\frac{2}{n^{2} \pi}\left[(-1)^{n}-1\right].\end{aligned}$$ Thus $$f(x)=\frac{\pi}{2}+\sum_{n=1}^{\infty} \frac{2}{n^{2} \pi}\left[(-1)^{n}-1\right] \cos n x.$$

Step-by-Step Solution

Answer

\(f(x) = \frac{\pi}{2} + \sum_{n=1}^{\infty} \frac{2}{n^{2}\pi}\left[(-1)^{n}-1\right] \cos nx.\)

Step 1: Understand Even Functions

An even function is symmetric around the y-axis, meaning \(f(x) = f(-x)\). When expanding an even function using Fourier series, only cosine terms appear because cosine is also an even function.

Step 2: Identify the Expansion Method

We use a cosine series expansion to represent an even function. The cosine series expansion is given by \(a_0\) and \(a_n\) terms, where \(a_0\) is the average value of the function over one period, and \(a_n\) are the coefficients for the cosine terms.

Step 3: Evaluate \(a_0\)

The coefficient \(a_{0}\) is calculated using the formula: \(a_{0} = \frac{2}{\pi} \int_{0}^{\pi} x \, dx\). Solving this integral, we get \(a_{0} = \frac{2}{\pi} \left[ \frac{x^2}{2} \right]_0^{\pi} = \pi\).

Step 4: Evaluate \(a_n\)

The coefficients \(a_n\) for \(n \geq 1\) are calculated using: \(a_{n} = \frac{2}{\pi} \int_{0}^{\pi} x \cos nx \, dx \). Solving this, we get \(a_{n} = \frac{2}{n^{2}\pi}\left[(-1)^{n}-1\right]\).

Step 5: Formulate the Cosine Series

We can now write the cosine series for \(f(x)\) as: \(f(x) = \frac{\pi}{2} + \sum_{n=1}^{\infty} \frac{2}{n^{2}\pi}\left[(-1)^{n}-1\right] \cos nx\).

Key Concepts

Even Function

An even function is one that has reflectional symmetry about the y-axis. This means for every point

• If you take the negative of any x-value (say -x), the function will yield the same y-value as it did for the positive x-value.
• In mathematical terms, an even function satisfies the condition: \( f(x) = f(-x) \).

Such functions include even-powered terms like \( x^2 \), \( x^4 \), and so on. Because of their symmetry, even functions often appear in problems involving oscillations and waveforms. When expanding even functions into a Fourier series, they exclusively use cosine terms. This is because cosine itself is an even function, meaning cosine at a positive angle is equal to cosine at the corresponding negative angle.

Cosine Series

The cosine series is a type of Fourier series used to represent even functions. It includes only cosine terms and constants. The cosine series expansion for even functions includes:

• \( a_0 \), which is the average value of the function over a complete cycle or period.
• \( a_n \) terms, which are the coefficients for each cosine term.

These coefficients help determine the weight of each cosine term in replicating the given function. The cosine series can effectively model periodic functions, especially those with even symmetry, across various applications such as signal processing, acoustics, and electromagnetic waves.

Series Expansion

A series expansion is a way to represent a mathematical function as an infinite sum of terms. For even functions, a cosine series expansion can be used. This entails using a combination of cosine functions of varying frequencies and amplitudes. The aim of such an expansion is to approximate a function with a series of simpler components. In the context of Fourier series:

• The base function or constant term is denoted by \( a_0 \).
• Subsequent terms comprise sums of cosine functions whose amplitudes are scaled by coefficients \( a_n \).
• These series allow complex periodic functions to be broken down, making them easier to analyze and manipulate.

This process is widely applicable in mathematical modeling, engineering, and physical sciences.

Integral Calculation

Integral calculations are essential in determining Fourier coefficients like \( a_0 \) and \( a_n \). The process involves calculating the area under the curve of a function over a specific interval. For our cosine series expansion, two main integrals need to be evaluated:

• The integral for \( a_0 \): \[ a_{0} = \frac{2}{\pi} \int_{0}^{\pi} x \, dx \] Solving this gives us \( a_{0} = \pi \), representing the average value.
• The integral for \( a_n \) when \( n \geq 1 \):\[ a_{n} = \frac{2}{\pi} \int_{0}^{\pi} x \cos n x \, dx \]After evaluating, we derive:\[ a_{n} = \frac{2}{n^{2}\pi}\left[(-1)^{n}-1\right]\]

These integrals enable the conversion of a given function into its Fourier cosine series, making it possible to explore complex periodic behaviors with simpler mathematical expressions.