Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Q10P (page 384)

Find the exponential Fourier transform of the given f(x)and write f(x)as a Fourier integral.

Short Answer

Expert verified

Answer

The exponential Fourier transform of the given function is gα2iπα2αa-sinαaand fxas a Fourier integral isfx=∫-∞∞αa-sinαaiπα2eiαxdα.

Step by step solution

01

Given information

The graph of the given function is as shown below.

In mathematical form, the function can be written as shown below.

fx=-2x+a,x∈-a,0-2x-a,x∈0,a.

The exponential Fourier transform of the function is to be found and the function is to be written as a Fourier integral.

02

The significance of Fourier transforms

The following are the formulas for the Fourier series transforms.

Here,is called the Fourier transform of.

03

Find the Fourier transform

Use the formula gα=22π∫-∞∞fxe-iαxdxto find the Fourier transform.

Solve further to obtain.

Now, writeas a Fourier integral using the formula.

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!

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

Represent each of the following functions (a) by a Fourier cosine integral, (b) by a Fourier sine integral. Hint: See the discussion just before theParseval’s theorem.

In each case, show that a particle whose coordinate is (a) x=Rez, (b) y=lmz is undergoing simple harmonic motion, and find the amplitude, period, frequency, and velocity amplitude of the motion.

z=2eiπt

Represent each of the following functions (a) by a Fourier cosine integral, (b) by a Fourier sine integral. Hint: See the discussion just before theParseval’s theorem

f(x)={1,0<x<π20,x>π2

Expand the same functions as in Problems 5.1 to 5.11 in Fourier series of complex exponentials einx on the interval(-π,π) and verify in each case that the answer is equivalent to the one found in Section 5.

In Problems 13to 16, find the Fourier cosine transform of the function in the indicated problem, and write f(x)as the Fourier integral [ use equation (12.15)]. Verify that the cosine integral forf(x) is the same as the exponential integral found previously.

16. Problem 11
See all solutions

Recommended explanations on Physics Textbooks

Rotational Dynamics

Read Explanation

Fluids

Read Explanation

Thermodynamics

Read Explanation

Kinematics Physics

Read Explanation

Geometrical and Physical Optics

Read Explanation

Electrostatics

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