Problem 15 - Calculus

Question

Suppose we have a periodic function \(f\) with period \(1 \mathrm{de}\) fined by \(f(x)=x\) for \(0 \leq x<1 .\) Find the fourth-degree Fourier polynomial for \(f\) and graph it on the interval \(0 \leq x<1 .\) IHint: Remember that since the period is not 2\pi, you will have to start by doing a substitution. Notice that the terms in the sum are not \(\sin (n x)\) and \(\cos (n x)\) but instead turn out to be \(\sin (2 \pi n x) \text { and } \cos (2 \pi n x) .]\)

Step-by-Step Solution

Answer

The Fourier series up to the fourth degree is: \( F_4(x) = \frac{1}{2} - \frac{1}{\pi} \sum_{n=1}^{4} \frac{\sin(2\pi n x)}{n} \).

Step 1: Define the function and period

We start with a periodic function defined by \(f(x) = x\) for \(0 \leq x < 1\). The period of this function is \(1\), not \(2\pi\).

Step 2: Perform substitution for standard Fourier series

Since the period is \(1\), substitute \(x\) with \(u = 2\pi x\) to use standard Fourier series concepts. Then the function becomes \(f(u) = \frac{u}{2\pi}\) for \(0 \leq u < 2\pi\).

Step 3: Determine Fourier coefficients

Calculate the Fourier coefficients using:- The constant term \(a_0 = \frac{1}{T}\int_0^T f(u) \, du\)- Cosine coefficients \(a_n = \frac{2}{T}\int_0^T f(u) \cos(nu) \, du\)- Sine coefficients \(b_n = \frac{2}{T}\int_0^T f(u) \sin(nu) \, du\)\, where \(T=2\pi\).Compute these integrals to obtain:- \(a_0 = \frac{1}{2}\)- \(a_n = 0\) for all \(n\)- \(b_n = -\frac{1}{\pi n}\) for \(n\geq 1\)

Step 4: Construct the Fourier series

The Fourier series for \(f\) up to the fourth degree is:\[ F_4(x) = \frac{1}{2} - \frac{1}{\pi} \sum_{n=1}^{4} \frac{\sin(2\pi n x)}{n} \]This represents the truncated Fourier series to degree \(4\).

Step 5: Graph the Fourier polynomial

Plot \(F_4(x)\) over the interval \(0 \leq x < 1\). This will show the approximation of the function \(f(x) = x\) using the first 4 sine terms of its Fourier series with period \(1\).

Key Concepts

Periodic Function

A periodic function is one that repeats its values at regular intervals or periods. In this exercise, we are dealing with a function defined by \( f(x) = x \) on the interval \( 0 \leq x < 1 \). This function repeats itself every period, which here is specified as \( 1 \).

Periodic functions are fundamental in the study of oscillations and waves. They capture the repeating nature of cyclic phenomena, such as the motion of waves or the swing of a pendulum.

When working with periodic functions, understanding the period is crucial, as it determines the function's behavior and helps in the calculation of Fourier series.

Fourier Coefficients

Fourier coefficients are the building blocks of Fourier series. They measure the contribution of each sine and cosine term to the overall series representation of a periodic function.

There are three types of Fourier coefficients:

• The Constant Term \(a_0\): Represents the average value of the function over one period.
• Cosine Coefficients \(a_n\): Measure the contribution of cosine terms. In this exercise, since the function is odd, these coefficients turn out to be zero.
• Sine Coefficients \(b_n\): Account for the sine terms, which are crucial for expressing odd functions like \(f(x) = x\).

To compute these coefficients, integrals over one period of the function are used, capturing the essence of how much each trigonometric function contributes to the shape of the original function.

Trigonometric Functions

Trigonometric functions such as sine and cosine are the backbone of Fourier series. They help represent periodic functions through oscillatory patterns.

In the context of Fourier series, these functions provide the orthogonal basis needed to decompose any periodic signal into sum components. For a function \( f(x) = x \), finding the Fourier series involves breaking it down into a sum of sines and cosines.

• Sine Functions: Useful for representing odd functions due to their symmetric properties.
• Cosine Functions: Typically used for even functions, but here they do not contribute due to the nature of \( f(x) \).

Each trigonometric function tries to "capture" certain oscillatory features of the function, making their overall sum a good approximation of the original function under the Fourier framework.

Graphing Fourier Polynomials

Graphing Fourier polynomials is a visualization exercise in approximation. By plotting the Fourier polynomial, you can see how well it approximates the original function within one period.

In this case, we graph the fourth-degree Fourier polynomial of \( f(x) = x \), which is given by: \[ F_4(x) = \frac{1}{2} - \frac{1}{\pi} \sum_{n=1}^{4} \frac{\sin(2\pi n x)}{n} \]

This polynomial includes the first four sine terms, each scaled by coefficients that were calculated earlier. Such a graph shows the attempt to match the linear function \( f(x) \) with a series of oscillatory terms.

• **Benefits:** Provides a clear visual of how Fourier series approximates real functions.
• **Observations:** The polynomial will show more fluctuation close to discontinuities due to Gibbs phenomenon, a common characteristic in Fourier series approximations.

Graphing helps students visualize and understand the convergence and limitations of Fourier polynomials, particularly when using a limited number of terms to approximate the original periodic function.