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Chapter 10: Problem 22

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. $$ f(x)=\left\\{\begin{array}{lr}{x+2,} & {-2 \leq x < 0,} \\ {2-2 x,} & {0 \leq x < 2}\end{array} \quad f(x+4)=f(x)\right. $$

Short Answer

Expert verified
Based on the given step-by-step solution, answer the following: 1. Sketch the graph for three periods of the given function. 2. Determine the Fourier series representation of the function. 3. Plot the partial sums \(s_m(x)\) for \(m=5,10\), and 20. 4. Describe how the Fourier series is converging.

Step by step solution

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01

Sketch the graph for three periods

First, let's evaluate the function for three periods. Since the period is 4, we will sketch the graph on the interval \([-8, 12)\) by just extending the function definition in intervals of length 4: - For \(-8 \le x < -6\), \(f(x+4) = f(x)\) which is \(x+6\). - For \(-6 \le x < -4\), \(f(x+4) = f(x)\) which is \(2-2(x+4)\). - For \(-4 \le x < -2\), \(f(x+4) = f(x)\) which is \(x + 4\). - For \(-2 \le x < 0\), \(f(x) = x+2\). - For \(0 \le x < 2\), \(f(x) = 2-2x\). - For \(2 \le x < 4\), \(f(x+4) = f(x)\) which is \(x\). - For \(4 \le x < 6\), \(f(x+4) = f(x)\) which is \(2-2(x-4)\). - For \(6 \le x < 8\), \(f(x+4) = f(x)\) which is \(x -4\). - For \(8 \le x < 10\), \(f(x+4) = f(x)\) which is \(2-2(x-8)\). Now sketch the graph of the function by plotting above pieces.
02

Find the Fourier Series

To calculate the Fourier series of a periodic function, we use the following formulas for coefficients: $$ a_0 = \frac{1}{L}\int_{-L}^L f(x)dx, \quad a_n = \frac{1}{L}\int_{-L}^{L}f(x)\cos\frac{n\pi x}{L}dx, \quad b_n = \frac{1}{L}\int_{-L}^{L}f(x)\sin\frac{n\pi x}{L}dx $$ where \(L\) is half the period (in this case \(L=2\)) and \(n=1,2,3,...\). For the given function, we need to calculate two separate integrals (one for each part of the piecewise definition) and sum them up: $$ a_0 = \frac{1}{2}\int_{-2}^0 (x+2)dx + \frac{1}{2}\int_{0}^{2}(2-2x)dx $$ $$ a_n = \frac{1}{2}\int_{-2}^0 (x+2)\cos\frac{n\pi x}{2}dx + \frac{1}{2}\int_{0}^{2}(2-2x)\cos\frac{n\pi x}{2}dx $$ $$ b_n = \frac{1}{2}\int_{-2}^0 (x+2)\sin\frac{n\pi x}{2}dx + \frac{1}{2}\int_{0}^{2}(2-2x)\sin\frac{n\pi x}{2}dx $$ Calculate the definite integrals for \(a_0, a_n,\) and \(b_n\), then construct the Fourier series as follows: $$ f(x) \approx \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\frac{n\pi x}{2} + b_n\sin\frac{n\pi x}{2}\right] $$
03

Plot \(s_m(x)\) for \(m=5,10\), and 20

To plot the partial sums \(s_m(x)\) for \(m=5,10\), and 20, truncate the Fourier series above at the desired value of \(m\): $$ s_m(x) = \frac{a_0}{2} + \sum_{n=1}^{m}\left[a_n\cos\frac{n\pi x}{2} + b_n\sin\frac{n\pi x}{2}\right] $$ Then, plot the function \(s_m(x)\) for each specified value of \(m\) on the interval \([-8, 12)\).
04

Describe how the Fourier series is converging

To analyze the convergence, compare the graphs of the partial sums \(s_m(x)\) with the graph of the original function \(f(x)\). As the value of \(m\) increases, observe whether the partial sum approximates the function more accurately. Also, notice if there are any overshoots (Gibbs phenomenon) or artifact oscillations in the approximations. Describe the convergence in terms of how the partial sums approach the function and any notable phenomena observed.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Periodic Function
A periodic function is a function that repeats its values in regular intervals or periods. In this problem, the given function, \(f(x)\), is defined for an interval \([-2, 2)\) and has a period of 4. This implies that the function values recur every 4 units along the x-axis.
You can imagine a periodic function as a loop that starts and ends at the same point. Whatever changes occur within one cycle (or period), they repeat identically in subsequent cycles. This characteristic is key when computing their Fourier series, as it leverages the function's repetitive nature.
  • Such functions help model real-world cyclical processes like sound waves or alternating current.
  • Understanding periodicity aids in predicting future values based on past and present data.
Recognizing and sketching periodic functions are foundational skills in learning how to develop their Fourier series representations.
Partial Sums
Partial sums refer to the finite series obtained by summing a limited number of terms of an infinite series. In our Fourier series exercise, partial sums \(s_m(x)\) approximate the function by summing the Fourier series up to \(m\) terms. This simplification provides a practical tool for analyzing the function's behavior without needing every term.
Consider it like capturing a sketch of your function's full portrait. The more terms you include (higher \(m\)), the closer you get to capturing the entire image. However, even with just a few terms (smaller \(m\)), you often get a reasonable approximation that can highlight significant features of the function.
  • Partial sums at different \(m\) values allow us to see how adding more terms affects accuracy.
  • They also let us visualize the gradual improvement in approximation of the original function.
Understanding partial sums is vital for appreciating how Fourier series representations develop and improve.
Convergence
Convergence in the context of Fourier series refers to how the partial sums \(s_m(x)\) of the series progressively approach the actual function \(f(x)\). This happens as \(m\), the number of terms in the sum, increases. The series is considered to converge if, as more terms are added, the partial sums get increasingly close to mimicking the periodic function throughout its domain.
Convergence is pivotal in determining whether the Fourier series provides a valid approximation of the function. When a series converges, any difference between the original function and its approximation diminishes toward zero. This is crucial to ensure that the Fourier series not only captures the general shape, but also the fine details of the original function.
  • Convergence is impacted by the function's smoothness. More abrupt changes can slow convergence.
  • We observe convergence practically by plotting partial sums and comparing them to the actual function.
By analyzing convergence, students can understand the effectiveness and limitations of using Fourier series for approximation.
Gibbs Phenomenon
The Gibbs phenomenon refers to an overshoot or oscillation that occurs near discontinuities in a function's Fourier series approximation. Whenever the original function features jumps, the Fourier series tends to exhibit noticeable "wiggles" near these points. This effect diminishes but never completely eradicates, no matter how many terms are added.
Imagine trying to draw a sharp corner with a rounded tool—some degree of rounding or overshoot is inevitable. In our problem, observing these overshoots and oscillations signifies the presence of the Gibbs phenomenon, particularly as the number of terms \(m\) increases.
  • This phenomenon is inherent due to the nature of Fourier series and does not imply errors in calculation.
  • Recognizing Gibbs overshoots helps in setting realistic expectations for the precision of Fourier approximations near discontinuities.
Understanding the Gibbs phenomenon prepares students for interpreting these common artifacts in practical Fourier analysis.

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