Logout Open In App
Open In App
Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 12: Problem 3

Which of the following functions of \(x\) could be represented by a Fourier series over the range indicated? (a) \(\tanh ^{-1}(x), \quad-\infty

Short Answer

Expert verified
Only \(\text{x sin}(1/x) \, \text{cyclically repeated over the range} \ -fty < x < fty\), can represent the Fourier series.

Step by step solution

01

- Understanding the Problem

Determine which of the given functions can be represented by a Fourier series over their respective specified ranges.
02

- Condition for Fourier Series

Recall that a function must be periodic to be represented by a Fourier series. Check if each function is periodic over the given range.
03

- Check \(\tanh^{-1}(x)\)

The function \(\tanh^{-1}(x)\) is not periodic over \(-fty < x < fty\). Therefore, it cannot be represented by a Fourier series.
04

- Check \(|\text{sin} x|^{-1/2}\)

The function \(|\text{sin} x|^{-1/2}\) is not periodic over \(-fty < x < fty\). Therefore, it cannot be represented by a Fourier series.
05

- Check \(\text{cos}^{-1}(\text{sin} 2 x)\)

The function \(\text{cos}^{-1}(\text{sin} 2 x)\) is not periodic over \(-fty < x < fty\). Therefore, it cannot be represented by a Fourier series.
06

- Check \(\text{x sin}(1/x)\)

Review the function \(\text{x sin}(1/x)\) over the interval given: \(-fty < x fty\). Since the function is not periodic over this interval, it also cannot be represented by a Fourier series.
07

- Identify the Cyclical Repetition

The function \(\text{x sin}(1/x)\) provided in the interval \(-fty < x fty\) is not given a specification that it repeats cyclically in this entire interval. Only the function \(\text{x sin}(1/x) \, \text{cyclically repeated over the range} \ -fty < x < fty\), can represent the Fourier series.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Periodic Functions
Periodic functions repeat their values in regular intervals or periods. The concept of periodicity is essential when dealing with Fourier series because a function must be periodic to be expanded in a Fourier series. Think of sine and cosine functions. They repeat their values every \(2\pi\). For example, \(\text{sin}(x)\) satisfies \(\text{sin}(x + 2\pi) = \text{sin}(x)\). This regular interval makes it possible to represent periodic functions through a sum of sines and cosines.
Inverse Hyperbolic Functions
Inverse hyperbolic functions like \(\tanh^{-1}(x)\) are the inverses of hyperbolic functions. Hyperbolic functions resemble trigonometric functions but are related to hyperbolas rather than circles. The function \(\tanh^{-1}(x)\) is defined for all \(x\) and does not repeat its values at regular intervals. This lack of periodicity means it cannot be represented by a Fourier series. If a function is not periodic, no combination of sine and cosine terms (which are inherently periodic) will adequately describe it everywhere.
Absolute Value Functions
An absolute value function represents the distance of a number from zero on the number line, ignoring its sign. For example, \(|\text{sin} x|^{-1/2}\) uses the absolute value of \(\text{sin}(x)\). Absolute value functions can sometimes be periodic themselves but adding a transformation, like raising to the power of \(-1/2\), often disrupts this periodicity. This is the case with \(|\text{sin} x|^{-1/2}\), making it unsuitable for Fourier series representation because there is no regular repeating pattern.
Inverse Trigonometric Functions
Inverse trigonometric functions, like \(\text{cos}^{-1}(\text{sin} 2 x)\), are the inverses of trigonometric functions. They provide the angle whose trigonometric function equals a given value. For example, \(\text{cos}^{-1}\) returns the angle whose cosine is a specific number. The function \(\text{cos}^{-1}(\text{sin} 2 x)\) does not exhibit periodic behavior over the entire real line, hence it cannot be expressed as a Fourier series. Periodicity is crucial for Fourier expansions, and inverse trigonometric functions often lose this periodic nature.
Cyclical Repetition
Cyclical repetition refers to repeating a function’s pattern over a specified interval. In the context of Fourier series, cyclical repetition ensures a function is periodic. For example, the function \(\text{x sin}(1/x)\) defined over \(-\frac{1}{\pi} < x \leq \frac{1}{\pi}\), when repeated over this interval, can be represented by a Fourier series. This is because, by defining the function as periodically repeating, we impose a structure that allows the use of sines and cosines to express it accurately over each cycle.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A string, anchored at \(x=\pm L / 2\), has a fundamental vibration frequency of \(2 L / c\), where \(c\) is the speed of transverse waves on the string. It is pulled aside at its centre point by a distance \(y_{0}\) and released at time \(t=0 .\) Its subsequent motion can be described by the series $$ y(x, t)=\sum_{n=1}^{\infty} a_{n} \cos \frac{n \pi x}{L} \cos \frac{n \pi c t}{L} $$ Find a general expression for \(a_{n}\) and show that only odd harmonics of the fundamental frequency are present in the sound generated by the released string. By applying Parseval's theorem, find the sum \(S\) of the series \(\sum_{0}^{\infty}(2 m+1)^{-4}\).

Find the complex Fourier series for the periodic function of period \(2 \pi\) defined in the range \(-\pi \leq x \leq \pi\) by \(y(x)=\cosh x\). By setting \(t=0\) prove that $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}+1}=\frac{1}{2}\left(\frac{\pi}{\sinh \pi}-1\right) $$

Show that Parseval's theorem for two functions whose Fourier expansions have. cosine and sine coefficients \(a_{n}, b_{n}\) and \(\alpha_{n}, \beta_{n}\) takes the form $$ \frac{1}{L} \int_{0}^{L} f(x) g^{*}(x) d x=\frac{1}{4} a_{0} \alpha_{0}+\frac{1}{2} \sum_{n=1}^{\infty}\left(a_{n} \alpha_{n}+b_{n} \beta_{n}\right) $$ (a) Demonstrate that for \(g(x)=\sin m x\) or \(\cos m x\) this reduces to the definition of the Fourier coefficients. (b) Explicitly verify the above result for the case in which \(f(x)=x\) and \(g(x)\) is the square-wave function, both in the interval \(-1 \leq x \leq 1\)

Consider the representation as a Fourier series of the displacement of a string lying in the interval \(0 \leq x \leq L\) and fixed at its ends, when it is pulled aside by \(y_{0}\) at the point \(x=L / 4\). Sketch the continuations for the region outside the interval that will (a) produce a series of period \(L\), (b) produce a series that is antisymmetric about \(x=0\), and (c) produce a series that will contain only cosine terms. (d) What are (i) the periods of the series in (b) and (c) and (ii) the value of the \({ }^{4} a_{0}\)-term' in (c)? (e) Show that a typical term of the series obtained in (b) is $$ \frac{32 y_{0}}{3 n^{2} \pi^{2}} \sin \frac{n \pi}{4} \sin \frac{n \pi x}{L} $$

Find the (real) Fourier series of period 2 for \(f(x)=\cosh x\) and \(g(x)=x^{2}\) in the range \(-1 \leq x \leq 1 .\) By integrating the series for \(f(x)\) twice, prove that $$ \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^{2} \pi^{2}\left(n^{2} \pi^{2}+1\right)}=\frac{1}{2}\left(\frac{1}{\sinh 1}-\frac{5}{6}\right) $$
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free