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Chapter 11: Problem 2

In Exercises \(1-8,\) find the Fourier series associated with the given functions. Sketch each function. $$ f(x)=\left\\{\begin{array}{ll}{1,} & {0 \leq x \leq \pi} \\ {-1,} & {\pi < x \leq 2 \pi}\end{array}\right. $$

Short Answer

Expert verified
The Fourier series is \( f(x) = \sum_{n=1, \text{ odd}}^{\infty} \frac{4}{n\pi} \sin(nx) \).

Step by step solution

01

Define the interval and period

The function is defined on the interval \([0, 2\pi]\) with a period \(T = 2\pi\). This format is ready for Fourier series expansion since the function repeats its pattern over \(0\leq x < 2\pi\). The fundamental frequency \(\omega_0\) is \(\frac{2\pi}{T} = 1\).
02

Determine the Fourier coefficients

The Fourier series for a periodic function \(f(x)\) is given by: \[f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos(nx) + b_n \sin(nx) \right)\]First, calculate \(a_0\): \[ a_0 = \frac{1}{T} \int_0^{2\pi} f(x) \, dx = \frac{1}{2\pi} \left( \int_0^{\pi} 1 \, dx + \int_{\pi}^{2\pi} (-1) \, dx \right) = \frac{1}{2\pi} (\pi - \pi) = 0 \]
03

Calculate the cosine coefficients (an)

\[ a_n = \frac{1}{\pi} \int_0^{2\pi} f(x) \cos(nx) \, dx \]Split the integral: \[ a_n = \frac{1}{\pi} \left( \int_0^{\pi} \cos(nx) \, dx - \int_{\pi}^{2\pi} \cos(nx) \, dx \right) \]Calculate:\[ \int \cos(nx) \, dx = \frac{\sin(nx)}{n} \]Thus:\[ a_n = \frac{1}{\pi} \left( \left. \frac{\sin(nx)}{n} \right|_0^{\pi} - \left. \frac{\sin(nx)}{n} \right|_{\pi}^{2\pi} \right) = 0 \] This results from: \[ \sin(n\pi) - \sin(0) = 0 \text{ and } \sin(2n\pi) - \sin(n\pi) = 0 \] Therefore, \(a_n = 0\) for all \(n\).
04

Calculate the sine coefficients (bn)

\[ b_n = \frac{1}{\pi} \int_0^{2\pi} f(x) \sin(nx) \, dx \]Split the integral:\[ b_n = \frac{1}{\pi} \left( \int_0^{\pi} \sin(nx) \, dx - \int_{\pi}^{2\pi} \sin(nx) \, dx \right) \]Calculate:\[ \int \sin(nx) \, dx = -\frac{\cos(nx)}{n} \]Thus:\[ b_n = \frac{1}{\pi} \left( \left. -\frac{\cos(nx)}{n} \right|_0^{\pi} + \left. \frac{\cos(nx)}{n} \right|_{\pi}^{2\pi} \right) \]This simplifies to:\[ b_n = \frac{2}{n\pi} \left( 1 - (-1)^n \right) \]If \(n\) is odd, \(1 - (-1)^n = 2 \) and if \(n\) is even, \(1 - (-1)^n = 0 \). Therefore,\[ b_n = \begin{cases} \frac{4}{n\pi}, & \text{if } n \text{ is odd} \ 0, & \text{if } n \text{ is even} \end{cases} \]
05

Construct the Fourier series

With \(a_0 = 0\), \(a_n = 0\) for all \(n\), and the above results for \(b_n\), the Fourier series becomes:\[ f(x) = \sum_{n=1, \text{ odd}}^{\infty} \frac{4}{n\pi} \sin(nx) \] This result reflects the series containing only sine terms for odd harmonics, which describes the periodic rectangular waveform of \(f(x)\).
06

Sketch the function

The function \(f(x)\) is a rectangular wave with amplitude 1 in the interval \([0, \pi]\) and amplitude -1 in the interval \((\pi, 2\pi]\). This rectangular waveform repeats every \(2\pi\). The Fourier series visually approximates this function by summing odd harmonics, visibly smoothing the transitions at \(x = \pi\).

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Key Concepts

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

Periodic Function
A periodic function is a type of function that repeats its pattern over a regular interval or cycle. For example, the function given in the exercise \[ f(x)=\left\{\begin{array}{ll}{1,} & {0 \leq x \leq \pi} \ {-1,} & {\pi < x \leq 2 \pi}\end{array}\right. \] exhibits periodic behavior with a period of \(2\pi\). This means that once the function reaches \(x = 2\pi\), it begins to repeat the same sequence from \(0\) to \(2\pi\) again. Key characteristics of periodic functions include:
  • The period length \(T\), which determines the interval after which the function repeats. In this case, \(T = 2\pi\).
  • The fundamental frequency \(\omega_0 = \frac{2\pi}{T}\), essential for computing the Fourier series.
Understanding periodic functions serves as the foundation for decomposing these functions into Fourier series, enabling analysis of their wave-like properties and components.
Exploring periodic behavior helps to tackle problems involving signal processing, waves, and other applications.
Sine and Cosine Coefficients
In Fourier analysis, the sine and cosine coefficients form the backbone of representing a periodic function as a series of sinusoidal components. These coefficients are determined through integration over one period of the function and are fundamental in constructing the Fourier series.

Cosine Coefficients \(a_n\)

The cosine coefficients \(a_n\) are calculated as follows:\[ a_n = \frac{1}{\pi} \int_0^{2\pi} f(x) \cos(nx) \, dx \]In the given exercise, these coefficients calculated to zero. This means that there is no cosine component contributing to the Fourier series representation of the rectangular wave, simplifying the overall expression.

Sine Coefficients \(b_n\)

The sine coefficients \(b_n\) are determined through:\[ b_n = \frac{1}{\pi} \int_0^{2\pi} f(x) \sin(nx) \, dx \]These coefficients are found mainly for odd \(n\), as the calculations show. They play a crucial role in defining only the sine terms that approximate the waveform, particularly highlighting the odd harmonics in the Fourier series expansion.
By evaluating these coefficients, we capture the unique frequencies and amplitudes that compose the original periodical signal.
Rectangular Wave
A rectangular wave is a type of non-sinusoidal waveform that alternates between two levels or amplitudes, creating a distinct step-like pattern. It's characterized by its abrupt transitions between these two states鈥�'high' and 'low'. In this exercise, the rectangular wave is defined mathematically by:\[ f(x)=\left\{\begin{array}{ll}{1,} & {0 \leq x \leq \pi} \ {-1,} & {\pi < x \leq 2\pi}\end{array}\right. \]Such a wave repeats itself every \(2\pi\), alternating between \(+1\) and \(-1\). This waveform is commonly used in digital signal processing and electronic circuits as it closely resembles binary signals.
The Fourier series allows us to express this wave as a sum of sine functions that approach the shape of a rectangular wave, smoothing out the jumps in its graph. Although this approach outputs a smoother version, it preserves essential frequency components indicative of the rectangular wave's nature.
Fourier Coefficients Calculation
The calculation of Fourier coefficients is a critical procedure in Fourier analysis, aimed at breaking down a complex waveform into simpler sinusoidal components.

Finding \(a_0\), \(a_n\), and \(b_n\)

  • \(a_0\), the average or DC component, computed as:\[ a_0 = \frac{1}{2\pi} \int_0^{2\pi} f(x) \, dx \]In this exercise, it yields zero, indicating that the function oscillates equally above and below the x-axis over one period, balancing to zero.
  • To find \(a_n\):\[ a_n = \frac{1}{\pi} \int_0^{2\pi} f(x) \cos(nx) \, dx \]The result \(a_n = 0\) confirms there are no cosine components.
  • For \(b_n\):\[ b_n = \frac{1}{\pi} \int_0^{2\pi} f(x) \sin(nx) \, dx \]Here, non-zero values appear only when \(n\) is odd, showcasing great significance in reconstructing the waveform with sine terms.

Constructing the Fourier Series

The Fourier series is thus constructed primarily from the odd sine components, representing:\[ f(x) = \sum_{n=1, \text{ odd}}^{\infty} \frac{4}{n\pi} \sin(nx) \]
This formula reveals how different sine waves sum to approximate the original rectangular waveform, essentially recreating its transition levels and unique temporal characteristics.

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Most popular questions from this chapter

The Cauchy condensation test says: Let \(\left\\{a_{n}\right\\}\) be a nonincreasing sequence \(\left(a_{n} \geq a_{n+1} \text { for all } n\right)\) of positive terms that converges to \(0 .\) Then \(\sum a_{n}\) converges if and only if \(\sum 2^{n} a_{2 n}\) converges. For example, \(\sum(1 / n)\) diverges because \(\Sigma 2^{n} \cdot\left(1 / 2^{n}\right)=\sum 1\) diverges. Show why the test works.

Compound interest, deposits, and withdrawals If you invest an amount of money \(A_{0}\) at a fixed annual interest rate \(r\) compounded \(m\) times per year, and if the constant amount \(b\) is added to the account at the end of each compounding period (or taken from the account if \(b<0 ),\) then the amount you have after \(n+1\) compounding periods is $$ A_{n+1}=\left(1+\frac{r}{m}\right) A_{n}+b $$ a. If \(A_{0}=1000, r=0.02015, m=12,\) and \(b=50\) , calculate and plot the first 100 points \(\left(n, A_{n}\right) .\) How much money is in your account at the end of 5 years? Does \(\left\\{A_{n}\right\\}\) converge? Is \(\left\\{A_{n}\right\\}\) bounded? b. Repeat part (a) with \(A_{0}=5000, r=0.0589, m=12,\) and \(b=-50 .\) c. If you invest 5000 dollars in a certificate of deposit (CD) that pays 4.5\(\%\) annually, compounded quarterly, and you make no further investments in the CD, approximately how many years will it take before you have \(20,000\) dollars? What if the CD earns 6.25\(\% ?\) d. It can be shown that for any \(k \geq 0\) , the sequence defined recursively by Equation \((1)\) satisfies the relation $$ A_{k}=\left(1+\frac{r}{m}\right)^{k}\left(A_{0}+\frac{m b}{r}\right)-\frac{m b}{r} $$ For the values of the constants \(A_{0}, r, m,\) and \(b\) given in part (a), validate this assertion by comparing the values of the first 50 terms of both sequences. Then show by direct substitution that the terms in Equation \((2)\) satisfy the recursion formula in Equation ( 1\()\) .

Find series solutions for the initial value problems in Exercises \(15-32\) . $$ y^{\prime \prime}-y=x, \quad y^{\prime}(0)=2 \text { and } y(0)=-1 $$

Use series to evaluate the limits in Exercises \(47-56\) $$ \lim _{t \rightarrow 0} \frac{1-\cos t-\left(t^{2} / 2\right)}{t^{4}} $$

Prove that if \(\left\\{a_{n}\right\\}\) is a convergent sequence, then to every positive number \(\epsilon\) there corresponds an integer \(N\) such that for all \(m\) and \(n\) , $$ m>N \text { and } n>N \Rightarrow\left|a_{m}-a_{n}\right|<\epsilon $$
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