Problem 1 - Advanced Engineering Mathematics

Question

Identifying \(p=2\) we have $$\begin{aligned} c_{n} &=\frac{1}{4} \int_{-2}^{2} f(x) e^{-i n \pi x / 2} d x=\frac{1}{4}\left[\int_{-2}^{0}(-1) e^{-i n \pi x / 2} d x+\int_{0}^{2} e^{-i n \pi x / 2} d x\right] \\ &=\frac{i}{2 n \pi}\left[-1+e^{i n \pi}+e^{-i n \pi}-1\right]=\frac{i}{2 n \pi}\left[-1+(-1)^{n}+(-1)^{n}-1\right]=\frac{1-(-1)^{n}}{n \pi i} \end{aligned}$$ and $$c_{0}=\frac{1}{4} \int_{-2}^{2} f(x) d x=0$$ Thus $$f(x)=\sum_{n=-\infty \atop n \neq 0}^{\infty} \frac{1-(-1)^{n}}{i n \pi} e^{i n \pi x / 2}$$

Step-by-Step Solution

Answer

Fourier coefficient is \(\frac{1-(-1)^{n}}{i n \pi}\) and \(c_0=0\).

Step 1: Understanding the Integral

The exercise involves finding the Fourier coefficients for a function on the interval \([-2, 2]\) with period \(p=2\). The given formula for \(c_n\) is the expression for Fourier coefficients, specifically focused on applying to the function \(f(x)\).

Step 2: Splitting the Integral

The integral for \(c_n\) is split into two parts: \([−2,0]\) and \([0,2]\). This is necessary because the function \(f(x) = -1\) on \([-2, 0]\) and \(f(x) = 1\) on \([0,2]\).

Step 3: Evaluate Each Integral

For the interval \([-2, 0]\), calculate the integral \(-\int_{-2}^{0} e^{-i n \pi x / 2} dx\). For \([0, 2]\), calculate the integral \(+\int_{0}^{2} e^{-i n \pi x / 2} dx\).

Step 4: Solve the Integrals

The integration yields terms involving exponential functions evaluated at bounds. The calculations lead to the expression: \(\frac{i}{2 n \pi}(-1 + e^{i n \pi} + e^{-i n \pi} - 1)\), which simplifies using the properties of exponentials.

Step 5: Simplify Expressions

Using the property \(e^{i n \pi} = (-1)^n\), the integrals simplify to \(\frac{i}{2 n \pi}(-1 + (-1)^n + (-1)^n - 1)\). Simplifying further gives \(\frac{1 - (-1)^n}{n \pi i}\).

Step 6: Calculating Constant Term

For \(c_0\), the constant term, calculate \(\frac{1}{4}\int_{-2}^{2} f(x) dx\). Given that \(-1 \) and \(1\) cancel over the symmetric interval, \(c_0 = 0\).

Step 7: Construct the Fourier Series

The function \(f(x)\) can now be represented as the Fourier series: \( f(x) = \sum_{n=-\infty, n eq 0}^{\infty} \frac{1 - (-1)^n}{i n \pi} e^{i n \pi x / 2}.\)This is a series summing over all \( eq 0\), using the coefficients we derived.

Key Concepts

Integral Calculus

Integral Calculus is an essential tool in understanding and solving the problem of finding Fourier coefficients in the given exercise. When dealing with Fourier series, you often need to compute integrals, as they are crucial for finding these coefficients. Integration helps in calculating the area under a curve, which can be used to understand the behavior of periodic functions over specific intervals.

In this exercise, integration is used to decompose a periodic function into sine and cosine components. The integral \[\int_{-2}^{2} f(x) e^{-i n \pi x / 2} dx\]is evaluated over two intervals, \([-2,0]\) and \([0,2]\) because the function changes its form in these sub-intervals. This approach highlights the use of integration for assessing piecewise functions, making integral calculus a powerful tool in mathematical analysis for determining Fourier coefficients.

Complex Exponentials

Complex Exponentials play a vital role in Fourier series analysis. Instead of using sine and cosine directly, complex exponentials \(e^{i n \pi x / 2}\) are employed to represent the periodic components of a function. This is because exponentials simplify the mathematics involved, making it easier to handle different operations such as multiplication and integration.

Complex exponentials have the property \(e^{i \theta} = \cos(\theta) + i\sin(\theta)\),which relates them directly to trigonometric functions. In the context of the exercise, \(c_n\) is expressed in terms of complex exponentials, allowing the series to be computed efficiently.

By using these exponentials, one can transform problems involving periodic functions into problems involving polynomial equations, which are often simpler to analyze and solve. This transformation underlines the significance of complex exponentials in engineering mathematics and other applied sciences.

Piecewise Functions

Piecewise functions are defined by different expressions depending on the interval of the variable input. They are particularly useful in modeling functions that have distinct behaviors in different intervals. In this exercise, the function \(f(x)\)is piecewise and defined as:

• \(f(x) = -1\) for \([-2,0]\)
• \(f(x) = 1\) for \([0,2]\)

This indicates that \(f(x)\)takes a constant value in each segment, which directly affects the calculation of the Fourier coefficients.

Understanding piecewise functions is crucial when you split integrals for computing Fourier coefficients, as each segment may require separate evaluations. Such functions appear frequently in real-world engineering problems, where conditions change over time or vary across different segments of a project or experiment.

Engineering Mathematics

In engineering mathematics, the concept of Fourier series and integrals is pivotal. They allow engineers to model and analyze systems that display periodic behavior, which is common in fields such as signal processing, electrical engineering, and vibrations analysis.

In the given exercise, representing a function using a Fourier series helps engineers break down complex oscillatory systems into simpler, calculable parts. This process aids in identifying the frequency components of the function, making it possible to understand and predict system behavior under different conditions.

Additionally, techniques like splitting integrals for piecewise functions, and using complex exponentials effectively, reflect the multidisciplinary utility of these mathematical approaches. By developing a strong foundation in these concepts, students and professionals can design, optimize, and innovate in various technological applications.