Problem 23 - Advanced Engineering Mathematics

Question

Expand the given function in an appropriate cosine or sine series. $$ f(x)=|\sin x|,-\pi

Step-by-Step Solution

Answer

The cosine series expansion of \(|\sin x|\) on \(-\pi < x < \pi\) is \( \frac{4}{\pi} + \sum_{n=1, \text{ odd}}^{\infty} \frac{4}{\pi} \frac{1}{1-n^2} \cos(nx) \).

Step 1: Understand the Function and Interval

The given function is \( f(x) = |\sin x| \) defined over the interval \(-\pi < x < \pi\). Our goal is to expand it as a series of either sine or cosine functions, which are periodic, over this interval.

Step 2: Determine the Symmetry of the Function

The function \( f(x) = |\sin x| \) is even because \( f(-x) = |\sin(-x)| = |\sin x| = f(x) \). Therefore, \( f(x) \) can be expanded as a cosine series because even functions have cosine series representations.

Step 3: Set Up the Cosine Series Formula

For an even function on the interval \(-\pi < x < \pi\), we use the cosine series:\[ f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos(nx) \]where the coefficients \( a_0 \) and \( a_n \) are given by:\[ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \, dx \]\[ a_n = \frac{2}{\pi} \int_{-\pi}^{\pi} f(x) \cos(nx) \, dx \]

Step 4: Calculate \( a_0 \)

Compute the constant term using the formula:\[ a_0 = \frac{1}{\pi} \int_{-\pi}^{\pi} |\sin x| \, dx \]Since \( |\sin x| \) is an even function, evaluate from 0 to \( \pi \) and double the result:\[ a_0 = \frac{2}{\pi} \int_{0}^{\pi} \sin x \, dx = \frac{2}{\pi} [-\cos x]_0^{\pi} = \frac{2}{\pi} [2] = \frac{4}{\pi} \]

Step 5: Calculate \( a_n \) for \( n \geq 1 \)

For \( n \geq 1 \), calculate the coefficients using:\[ a_n = \frac{2}{\pi} \int_{-\pi}^{\pi} |\sin x| \cos(nx) \, dx \]Due to symmetry of \( |\sin x| \), use:\[ a_n = \frac{4}{\pi} \int_{0}^{\pi} \sin x \cos(nx) \, dx \]Use integration by parts or known integral results to determine that \( a_n = \frac{4}{\pi} \left( \frac{1+(-1)^{n+1}}{1-n^2} \right) \) for odd \( n \), and \( a_n = 0 \) for even \( n \).

Step 6: Verify and Write the Final Cosine Series

Combine \( a_0 \) and \( a_n \) into the cosine series:\[ f(x) = \frac{4}{\pi} + \sum_{n=1, \text{ odd}}^{\infty} \frac{4}{\pi} \frac{1}{1-n^2} \cos(nx) \]This series representation satisfies the given function over the interval \(-\pi < x < \pi\).

Key Concepts

Trigonometric Series

Trigonometric series are mathematical tools used to represent periodic functions through sums of sine and cosine functions. This method is versatile and forms the backbone of many applications in mathematics, engineering, and physics. When a function is expanded in terms of trigonometrical components, it provides insights into the function's behavior over a definite period.

The general form of a trigonometric series is:

• For a full Fourier Series expansion, which represents a function over an interval, the series is given as: \[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right) \]
• The coefficients \(a_n\) and \(b_n\) tell us how much of each cosine or sine wave is needed to construct the function.

With trigonometric series, the goal is to break down complex periodic functions into simpler sinusoidal components, which are easier to analyze and interpret. Understanding this concept opens up a world of mathematical analysis that is essential for solving real-world problems involving periodicity.

Cosine Series

A cosine series is a special type of trigonometric series where only cosine functions are used to approximate an even function. An even function satisfies the symmetry condition:

• \(f(x) = f(-x)\).

Because of this symmetry, the sine terms in the Fourier series expansion, which are odd functions, drop out, leaving only cosine terms.

The representation of a cosine series for an even function over \(-\pi < x < \pi\) is:

• \[f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos(nx)\]

The coefficient \(a_0\) represents the average value of the function over one period, and the coefficients \(a_n\) provide the amplitude of each cosine component in the series.

The use of cosine series is prevalent in cases where the function exhibits even symmetry, making it a straightforward and effective approach for analysis.

Even Function Expansion

Even functions have a particular property where the function is symmetric about the y-axis. This symmetry is defined mathematically as:

• \(f(-x) = f(x)\) for all \(x\) in the domain.

Expanding even functions using cosine series leverages this symmetry and simplifies the expression by using only cosine terms.

In the context of Fourier series, for an even function, the cosine series representation becomes:

• \[f(x) = a_0 + \sum_{n=1}^{\infty} a_n \cos(nx)\]

To find the coefficients \(a_0\) and \(a_n\), evaluate the integrals over the interval, taking advantage of the symmetry to often simplify the calculations. This approach significantly reduces the complexity involved in deriving a series expansion for functions that exhibit even property.

Utilizing cosine series for even function expansions helps in efficiently solving problems across various scientific fields, leading to simpler computational models and improved understanding of the underlying phenomena.