Question

Find the mixed Fourier cosine series. $$ f(x)=x^{2} ; \quad[0, L] $$

Short answer

Answer: The mixed Fourier cosine series representation of the function is given by: $$ f(x) = \frac{2L^2}{3} + \sum_{n=1}^{\infty} \frac{4}{n^2\pi^2}\left[1 - \cos(n\pi)\right] \cos\left(\frac{n\pi x}{L}\right) $$

Step-by-step solution

Define the given function and interval

Given function \(f(x) = x^2\) within the interval \((0, L)\).

Calculate A_0 (Constant term)

To find the constant term \(A_0\), we will use the following formula: $$ A_0 = \frac{2}{L} \cdot \int_0^L f(x) \, dx $$ Substitute \(f(x) = x^2\) into the formula: $$ A_0 = \frac{2}{L} \cdot \int_0^L x^2 \, dx $$ Now, integrate \(x^2\) with respect to \(x\): $$ A_0 = \frac{2}{L} \cdot \frac{x^3}{3} \Big|_0^L = \frac{2}{L} \cdot \frac{L^3}{3} $$ Finally, simplify the expression for \(A_0\): $$ A_0 = \frac{2L^2}{3} $$

Calculate A_n (Cosine terms coefficients)

To find the coefficients \(A_n\) for \(n \ge 1\), we will use the following formula: $$ A_n = \frac{2}{L} \cdot \int_0^L f(x) \cdot \cos\left(\frac{n \pi x}{L}\right) \, dx $$ Substitute \(f(x) = x^2\) into the formula: $$ A_n = \frac{2}{L} \cdot \int_0^L x^2 \cdot \cos\left(\frac{n \pi x}{L}\right) \, dx $$ Using integration by parts (with \(u = x^2\) and \(dv = \cos\left(\frac{n \pi x}{L}\right)dx\)), or looking up the integral in a table, we get: $$ A_n = \frac{2}{n^2 \pi^2} \cdot [(L^2 - \frac{2L}{n\pi}) \sin(\frac{n\pi L}{L}) + 2\cos(\frac{n\pi L}{L}) - 2] $$ Since \(\sin(\frac{n\pi L}{L})= \sin(n\pi)=0\) $$ A_n = \frac{4}{n^2\pi^2}\left[1 - \cos(n\pi)\right] $$

Write the mixed Fourier cosine series

Now, we can write the mixed Fourier cosine series for the given function \(f(x) = x^2\) in the interval \([0, L]\) using the coefficients \(A_0\) and \(A_n\): $$ f(x) = \frac{2L^2}{3} + \sum_{n=1}^{\infty} \frac{4}{n^2\pi^2}\left[1 - \cos(n\pi)\right] \cos\left(\frac{n\pi x}{L}\right) $$

Key concepts

Fourier series

The Fourier series is a powerful mathematical tool used to express a function as a sum of sine and cosine terms. Essentially, it breaks down complex periodic functions into simpler, periodic components. This decomposition helps us analyze the function in terms of frequency content, making it particularly useful in engineering and physics.
This theory is rooted in the idea that any periodic function can be reconstructed by summing an appropriate set of sine and cosine terms. Each term in the series corresponds to a specific frequency, amplitude, and phase. The basic form of a Fourier series for a function \( f(x) \) is:

• Constant term
• Sine and cosine terms for harmonics

By employing Fourier series, we can solve differential equations, analyze signals, and perform various calculations with functions that are periodic in nature.

integration by parts

Integration by parts is a technique used to solve integrals where the standard methods of integration don't apply easily. It's particularly useful when dealing with the product of two functions, as it transforms the integral into simpler parts. The formula for this technique is given by:\[\int u \, dv = uv - \int v \, du\]Here, you choose one part of the integrand as \( u \) and the other as \( dv \). Differentiate \( u \) to get \( du \) and integrate \( dv \) to get \( v \).
In the context of finding cosine coefficients in a Fourier series, integration by parts is used to simplify complex product integrals. For example, in our exercise, \( u = x^2 \) was chosen to simplify the integral involving cosine, breaking it down into manageable parts.

cosine coefficients

Cosine coefficients are an essential part of determining the cosine terms in a Fourier series. For a given function \( f(x) \), the cosine coefficients \( A_n \), including the constant term \( A_0 \), are calculated using integration. This process requires evaluating integrals involving products of \( f(x) \) and cosine functions:

• \( A_0 \) is the average value of the function over its period
• \( A_n \) represents how much of the cosine component is present

The cosine terms capture the even part of the function, which is crucial in applications like signal processing and vibrations analysis.
In computing \( A_n \), you often use symmetries and properties of cosine, like \( \cos(n\pi) \), which simplify calculations. These coefficients help construct the complete Fourier cosine series representation of a function.

trigonometric series

A trigonometric series is a series of terms that involve the trigonometric functions sine or cosine to describe periodic phenomena. In the context of Fourier series, it is the collection of sine and cosine terms that represent a function.

• Sine series capture odd behaviors
• Cosine series capture even behaviors

Understanding these series is essential for manipulating and interpreting complex waveforms and signals. They allow for the approximation of functions that may not be easily expressed as simple polynomials.
The mixed Fourier cosine series used in our exercise showcases how a trigonometric series can effectively approximate a quadratic function \( f(x) = x^2 \) over the interval \([0, L]\), providing insights into its harmonic structure.