/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Find the one-sided Fourier cosin... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 8

Find the one-sided Fourier cosine transform of the function \(a /\left(b^{2}+t^{2}\right)\).

Short Answer

Expert verified
The Fourier Cosine Transform of the function \( g(t) = \frac{a}{b^{2}+t^{2}} \) is \( G(\omega) = \sqrt{\frac{2}{\pi}} \left(\frac{a atan(\omega b)}{b}\right) \).

Step by step solution

01

- Definition of one-sided Fourier cosine transform

The one-sided Fourier cosine transform of a function \(g(t)\) is defined as \[G(\omega) = \sqrt{\frac{2}{\pi}} \int_{0}^{\infty} \cos(\omega t) g(t) dt\]. The function to be transformed is given as \(g(t) = \frac{a}{b^{2}+t^{2}}\).
02

- Substitution

First, let's represent the integral in a more convenient form. In the integral represent \(b^{2}+t^{2}\) as \(b^{2}(1 + (t/b)^{2})\), so the function g(t) will reform into \(g(t) = \frac{a/b^2}{1+(t/b)^2}\). The substitution \(x=t/b, dx=dt/b\), gives \[G(\omega) = \sqrt{\frac{2}{\pi}} \left(\frac{a}{b}\right) \int_{0}^{\infty} \cos\left(\omega x b\right) \frac{dx}{1+x^2}\]
03

- Completing the integral

We can recognize standard integral \(\int dx/(1+x^2) = atan(x) + C\) and it will give \[G(\omega) = \sqrt{\frac{2}{\pi}} \left(\frac{a}{b}\right) [atan(\omega b) + C]\]. The constant of integration \(C\) vanishes as the integral is from 0 to \(\infty\).
04

- Conclusion

The Fourier Cosine Transform of the function is \[G(\omega) = \sqrt{\frac{2}{\pi}} \left(\frac{a atan(\omega b)}{b}\right) \].

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Integral Calculus
Integral calculus is a fundamental branch of mathematics. It deals with integrals and their properties. In simpler terms, integration is the process of calculating the area underneath a curve. This is akin to adding up a series of infinitely small areas to obtain a total.
  • Integration helps in solving problems related to area, volume, displacement, etc.
  • In the context of the Fourier cosine transform, integration is necessary to determine the frequency components of a continuous signal.
To solve for a Fourier cosine transform, you need to set up an integral that represents the area under the function you're examining. In our exercise, we substituted values to simplify the integral. This is a common technique in calculus used to transform the problem into an easier one to handle.
Fourier Analysis
Fourier analysis is a powerful method in mathematical analysis. It breaks down complex signals into sine and cosine waves, which are easier to work with. You can think of it as a musical chord being broken into individual notes.
  • The Fourier cosine transform, in particular, focuses on expressing a signal using cosine waves, which are even functions.
  • This is important in systems where signals return to a baseline, such as in heat conduction problems.
In our problem, the Fourier cosine transform is applied to a specific function to better understand its frequency components. By using cosine functions in the transform process, it's possible to describe even characteristics of the function over a half-space (from 0 to infinity).
Mathematical Transforms
Mathematical transforms are operations that convert a function from one form to another. This conversion facilitates easier analysis and solutions.
  • They are crucial in turning real-world signal problems into solvable mathematical ones.
  • Transforms like the Fourier are used in fields from engineering to computer science.
The Fourier cosine transform is one type of mathematical transform. It efficiently turns a time domain function into a frequency domain representation. This change lets us understand how different frequencies contribute to the overall behavior. In solving our specific exercise, we effectively shifted the perspective from time-based observations to frequency-based insights.

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

Find the inverse Laplace transform of \(1 /\left(s^{2}-a^{2}\right)\).

Find the one-sided Fourier sine transform of the function \(a e^{-b x}\).

Show that \(\mathcal{L}\\{t \cos (k t)\\}=\left(s^{2}-k^{2}\right) /\left(s^{2}+k^{2}\right)^{2}\).

Find the Fourier series to represent the function $$ f(x)=\left\\{\begin{array}{r} -1 \text { if }-L \prec x<-L / 2 \\ 1 \text { if }-L / 2 \prec x

Find the Fourier transform of the function exp \(\left(-\left(x-x_{0}\right)^{2}\right)\)
See all solutions

Recommended explanations on Physics Textbooks

Quantum Physics

Read Explanation

Fields in Physics

Read Explanation

Space Physics

Read Explanation

Magnetism

Read Explanation

Electricity and Magnetism

Read Explanation

Fluids

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.