Question

In each of Problems 27 through 30 a function is given on an interval \(0

Short answer

#Sample Short Answer# To find the Fourier cosine and sine series for the given function \(f(x) = 3-x\) on the interval \(0 < x < 3\), we first sketch the even and odd extensions of \(f(x)\), which are found to be: $$ g(x)=\left\{ \begin{array}{ll} 3-x, & 0<x\le 3 \\ x-3, & 3<x<6 \end{array} \right., \quad h(x)=\left\{ \begin{array}{ll} 3-x, & 0<x\le 3 \\ x-9, & 3<x<6 \end{array} \right. $$ Then, we compute the Fourier cosine series coefficients: $$ a_0 = 1 \\ a_n = -\frac{6\sin(n\pi)}{n\pi} $$ and the Fourier sine series coefficients: $$ b_n = 6\frac{n\pi\cos(n\pi)+3\sin(n\pi)-n\pi}{n^2\pi^2} $$ Thus, the Fourier cosine and sine series are given by: $$ G(x) = 1- \sum_{n=1}^{\infty} \frac{6\sin(n\pi)}{n\pi} \cos\left(\frac{n\pi x}{3}\right), \\ H(x) = \sum_{n=1}^{\infty} 6\frac{n\pi\cos(n\pi)+3\sin(n\pi)-n\pi}{n^2\pi^2}\sin\left(\frac{n\pi x}{3}\right) $$ We plot the partial sums of each series, with different sums taken up to some integer \(N\). Finally, to investigate the dependence of the maximum error on \(n\), we compute the error function for each series and investigate its maximum over the interval \([0,3]\) by plotting maximum errors as a function of \(n\).

Step-by-step solution

1. Sketching Even and Odd Extensions#g_tag_content# To sketch the even and odd extensions of the given function, first recall that the even extension \(g(x)\) of \(f(x)\) is defined as $$ g(x) = \left\{ \begin{array}{ll} f(x) & \text{if} \quad 0 < x \le L\\ f(2L - x) & \text{if} \quad L < x < 2L \end{array} \right. $$ Similarly, the odd extension \(h(x)\) is defined as $$ h(x) = \left\{ \begin{array}{ll} f(x) & \text{if} \quad 0 < x \le L\\ -f(2L - x) & \text{if} \quad L < x < 2L \end{array} \right. $$ For our given function, with \(L = 3\) and \(f(x) = 3-x\), $$ g(x)=\left\{ \begin{array}{ll} 3-x, & 0

2. Fourier Cosine and Sine Series#g_tag_content# Denote the Fourier cosine series of the even extension \(g(x)\) as \(G(x)\) and the Fourier sine series as \(H(x)\) for the odd extension \(h(x)\). To find these series, we are going to compute the Fourier coefficients. The Fourier cosine series coefficients are given by $$ a_0 = \frac{2}{L} \int_0^L f(x) dx \\ a_n = \frac{2}{L} \int_0^L f(x) \cos\left(\frac{n\pi x}{L}\right) dx $$ and the Fourier sine series coefficients are $$ b_n = \frac{2}{L} \int_0^L f(x) \sin\left(\frac{n\pi x}{L}\right) dx $$ For our given function, we have $$ a_0 = \frac{2}{3} \int_0^3 (3-x) dx = 1 \\ a_n = \frac{2}{3} \int_0^3 (3-x) \cos\left(\frac{n\pi x}{3}\right) dx = -\frac{6\sin(n\pi)}{n\pi}\\ b_n = \frac{2}{3} \int_0^3 (3-x) \sin\left(\frac{n\pi x}{3}\right) dx = 6\frac{n\pi\cos(n\pi)+3\sin(n\pi)-n\pi}{n^2\pi^2} $$ Thus, the Fourier cosine and sine series are $$ G(x) = 1- \sum_{n=1}^{\infty} \frac{6\sin(n\pi)}{n\pi} \cos\left(\frac{n\pi x}{3}\right), \\ H(x) = \sum_{n=1}^{\infty} 6\frac{n\pi\cos(n\pi)+3\sin(n\pi)-n\pi}{n^2\pi^2}\sin\left(\frac{n\pi x}{3}\right) $$

3. Plot Partial Sums#g_tag_content# We can plot the partial sums of the Fourier cosine and sine series by taking the sum up to some integer \(N\). For instance, we could choose \(N=5\) and \(N=10\) as the first few partial sums.

4. Maximum Error Dependence on \(n\)#g_tag_content# To investigate the dependence of the maximum error on \(n\), we need to compute the error function of each series, which is given by the difference between the original function and the Fourier approximation: For the cosine series, $$ E_{G}(x,n) = |f(x) - G_N(x)|, $$ where \(G_N(x)\) is the N-th partial sum of the Fourier cosine series. For the sine series, $$ E_{H}(x,n) = |f(x) - H_N(x)|, $$ where \(H_N(x)\) is the N-th partial sum of the Fourier sine series. The maximum error for each series is found by taking the maximum of the error function over the interval \([0,L]\) for a given value of \(n\). By plotting the maximum error values as a function of \(n\), we can investigate the dependence of the maximum error on \(n\).

Key concepts

Even and Odd Extensions

When dealing with Fourier series, it's important to comprehend the concepts of even and odd extensions, as they form the basis for developing the Fourier cosine and sine series respectively. An even extension of a function means taking the original function defined on a given interval and extending it to the rest of the real line so that the extension is an even function, which means it is symmetric with respect to the y-axis. In mathematical terms, for a function defined on the interval \(0
On the other hand, the odd extension implies extending the function so that the result is symmetric with respect to the origin, meaning \(h(x) = -h(-x)\). This has direct implications on the Fourier series representation as even functions only have cosine terms (Fourier cosine series), while odd functions only have sine terms (Fourier sine series). For the given function \(f(x)=3-x\), the even extension \(g(x)\) would mirror the function on the y-axis for negative values, whereas the odd extension \(h(x)\) would mirror and flip it across the origin.

The visual representation of these extensions helps students understand how a non-periodic function can be converted into a periodic one, which can then be analyzed using Fourier series. Real-world applications of this knowledge abound, such as in signal processing where even and odd components of signals need to be separately analyzed.

Fourier Cosine Series

The Fourier cosine series represents an even periodic extension of a function using a sum of cosine functions with different frequencies. The coefficients of these series, \(a_0, a_n\), are determined by integrating the product of the original function and cosine terms over the original interval. For an even function \(g(x)\) defined on \(0
For the provided function \(f(x)=3-x\), the cosine series helps us describe the even periodic extension beyond the original interval. Interestingly, in this particular example, due to the symmetries of the cosine function, some of the coefficients, specifically those with \(a_n\), turn out to be zero as \(\sin(n\pi)\) becomes zero for integers \(n\). This simplification is a great example of how symmetry in a function can lead to a more compact Fourier representation. It demonstrates how the Fourier analysis breaks down complex periodic patterns into simpler harmonic components.

Fourier Sine Series

The Fourier sine series is similar to the cosine series but is exclusively used for representing odd periodic extensions of a function with sine functions. The coefficients \(b_n\) are obtained by integrating the product of the original function and sine functions over the interval. This sine series is significant when analyzing waveforms, vibrations, and heat distribution problems where the function's odd symmetry can be seen in the physical phenomena.

Using our example \(f(x)=3-x\), we compute the coefficients \(b_n\) through integration involving sine terms, which leads to a series of sine waves that capture the essence of the odd function's behavior. This kind of analysis is invaluable because certain physical conditions, such as those seen in mechanical systems or electrical circuits, naturally exhibit odd function characteristics and hence can be readily described using sine series.

Partial Sums

Partial sums are essential in visualizing how a Fourier series approximates a function as we gradually include more terms from the series. A partial sum \(S_N(x)\) of a Fourier series consists of a finite number of terms, up to the \(N\)-th term. As we increase \(N\), the approximation should ideally get closer to the actual function, albeit this behavior can vary depending on the function and the nature of the convergence.

In practice, by computing the first few partial sums of the Fourier series for our example \(f(x)=3-x\), students can observe how these sums approximate the function in different ways—relatively smoother with the cosine series and more abrupt with the sine series. These plots serve as an empirical bridge between abstract mathematical concepts and tangible visual representation, giving students a better intuition for mathematical theories.

Maximum Error

The maximum error is a measure of how well a Fourier series approximates the original function within a certain interval. As we use more terms in the partial sum, we generally expect the error to decrease. The error for each partial sum, especially the maximum error over the interval of interest \(0
For the function \(f(x)=3-x\), investigating the maximum error associated with the cosine and sine series as \(n\) increases can provide insights into convergence rates. Typically, the faster the maximum error decreases with increasing \(n\), the better the series converges. Understanding this relationship is crucial not just academically, but also in real-world scenarios where one has to balance the need for accuracy with computational or measurement limitations.