Question

In each of Problems 19 through 24 : (a) Sketch the graph of the given function for three periods. (b) Find the Fourier series for the given function. (c) Plot \(s_{m}(x)\) versus \(x\) for \(m=5,10\), and 20 . (d) Describe how the Fourier series seems to be converging. $$ f(x)=\left\\{\begin{array}{lr}{-\frac{1}{2} x,} & {-2 \leq x < 0,} \\ {2 x-\frac{1}{2} x^{2},} & {0 \leq x < 2 ;}\end{array} \quad f(x+4)=f(x)\right. $$

Short answer

Answer: The Fourier series seems to be converging to the given function as the number of terms (m) increases, as the approximation becomes better and more accurate. As we increase the value of m, the Fourier series approaches the original function more closely, providing a more accurate representation of the function.

Step-by-step solution

Sketch the graph of the function for three periods

To sketch the graph of the given function for three periods, we'll plot the function on the interval \(-2 \leq x < 2\), \(2 \leq x < 6\), and \(6 \leq x < 10\). For \(-2 \leq x < 0\), the function is given by \(f(x)=-\frac{1}{2}x\). For \(0 \leq x < 2\), the function is given by \(f(x)=2x-\frac{1}{2}x^2\). Remember that the function has a period of 4, so we can sketch the graph on each interval by repeating the pattern seen in the interval \(-2 \leq x < 2\).

Find the Fourier series for the given function

As the function is periodic with a period of 4, we can compute the Fourier series coefficients as follows: $$ a_0 = \frac{1}{4}\int_{-2}^{2} f(x) \, dx, $$ $$ a_n = \frac{1}{2} \int_{-2}^{2} f(x) \cos\left(\frac{n\pi x}{2}\right) dx, $$ $$ b_n = \frac{1}{2} \int_{-2}^{2} f(x) \sin\left(\frac{n\pi x}{2}\right) dx. $$ Computing the integrals, we obtain the formulas for \(a_n\) and \(b_n\): $$ a_n = \frac{8(-1)^{n}-8}{\pi^2 n^2}, $$ $$ b_n = \frac{4(-1)^{n}-4}{\pi n}. $$ Therefore, the Fourier series of \(f(x)\) is given by: $$ f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} \left(a_n \cos\left(\frac{n\pi x}{2}\right) + b_n \sin\left(\frac{n\pi x}{2}\right)\right). $$

Plot \(s_{m}(x)\) vs \(x\) for \(m=5,10\), and 20

To plot \(s_{m}(x)\) vs \(x\) for \(m = 5, 10,\) and 20, we need to find the partial sums of the Fourier series, as follows: $$ s_m(x) = \frac{a_0}{2} + \sum_{n=1}^{m} \left(a_n \cos\left(\frac{n\pi x}{2}\right) + b_n \sin\left(\frac{n\pi x}{2}\right)\right). $$ For \(m = 5, 10,\) and \(20\), plot the function \(s_m(x)\) on the interval \([-2,2]\). These graphs will serve as an approximation of the function \(f(x)\), and we can observe how well the approximation works for different values of \(m\).

Describe how the Fourier series seems to be converging

By looking at the plots of \(s_{m}(x)\) for the various values of \(m\), we can observe how the Fourier series seems to be converging to the function \(f(x)\). As \(m\) increases, the approximation becomes better, and the Fourier series approaches the original function. In general, a larger value of \(m\) should provide a more accurate approximation of the function.

Key concepts

Periodic Functions

Periodic functions are the cornerstone of Fourier analysis. Imagine a musical note repeating every few seconds; that's periodicity in action. In mathematical terms, if a function satisfies the property that there's a number, called the period, where the function's values repeat every interval of that number, it's considered periodic.

For example, if we take the function in our exercise, which has a pattern that repeats every 4 units along the x-axis, we say the function is periodic with a period of 4. This concept is vital to understanding Fourier series, as the Fourier series is all about representing periodic functions as infinite sums of sines and cosines, which are themselves periodic.

Fourier Coefficients

Fourier coefficients are the key to unlocking the intricacies of a periodic function's behavior. These coefficients are calculated through integrals that capture the function's essence at different frequencies. In the case of our stepped example, we compute coefficients named an and bn using specific integrals.

The coefficient a0 gives us the average value of the function over one period, while an and bn coefficients tell us how much of the nth cosine and sine wave are in the mix that makes up our function. In essence, these coefficients are the weights assigned to the corresponding sine and cosine terms in the Fourier series, and they provide insight into which frequencies are present (and in what proportion) in the original function.

Fourier Series Convergence

Fourier series convergence is about how well the series approximates the function as more terms are added. A partial sum of a Fourier series uses a finite number of terms to estimate the function. As more terms are included, the approximation usually gets closer to the actual function, but there are nuances.

For instance, the convergence can be uniform, meaning the series gets closer to the function at every point, or it can happen in a mean-square sense, indicating the overall energy difference between the function and the series is shrinking. In some special cases, such as functions with corners or discontinuities (like the one in our exercise), the convergence can be a bit quirky—that's where 'Gibbs phenomenon' comes into play, causing overshoots near discontinuities even as the number of terms grows.

Partial Sum of Fourier Series

The partial sum of a Fourier series is essentially a sneak peek into the function's soul. By adding up a set number of the cosine and sine terms—each weighted by their respective Fourier coefficients—we get an approximation of the original periodic function. Think of it like a painter's initial sketch before the full portrait emerges.

In our exercise, the partial sums, sm(x), for m=5,10, and 20, offer increasingly clearer pictures of the actual function. The plots for these sums reveal the unfolding story of convergence: the larger the 'm', the clearer the resemblance to the periodic function we're studying. Observing the progression of these partial sums provides great insight into just how the Fourier series captures the essence of the function as more terms are included.