/*! 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 25 Give simple examples of function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 25

Give simple examples of functions (defined for all \(x \geq 0\) ) that have no Laplace transform.

Short Answer

Expert verified
Functions like \( f(t) = e^{t^2} \) have no Laplace transform because they grow too fast.

Step by step solution

01

Understanding Laplace Transform

The Laplace transform of a function \( f(t) \) is defined as \( L\{f(t)\} = \int_0^\infty e^{-st}f(t) \, dt \). For the transform to exist, this integral must converge for some value of \( s \).
02

Condition for Laplace Transform Existence

For a function \( f(t) \) to have a Laplace transform, \( e^{-st} f(t) \) must decay to zero as \( t \) approaches infinity. This often requires that \( f(t) \) grows slower than \( e^{st} \).
03

Consider Functions with Rapid Growth

Consider exponential functions like \( f(t) = e^{at} \) with \( a > 0 \). The term \( e^{-st} e^{at} = e^{(a-s)t} \) will not decay to zero as \( t \) approaches infinity if \( a \geq s \).
04

Function Example

For an example, take \( f(t) = e^{t^2} \). Here, the function grows faster than any exponential \( e^{st} \) because \( e^{t^2} \) dominates \( e^{st} \) as \( t \to \infty \), causing \( \int_0^\infty e^{-st}e^{t^2} \, dt \) to diverge.
05

Validate Non-existence of Laplace Transform

The integral \( \int_0^\infty e^{t^2-st} \, dt \) diverges because \( e^{t^2-st} \rightarrow \infty \) as \( t \to \infty \) for any finite \( s \). Therefore, \( f(t) = e^{t^2} \) does not have a Laplace transform.

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.

Convergence
The term *convergence* is crucial to understanding the concept of the Laplace transform. In mathematical terms, a series or an integral converges if it approaches a finite limit as its terms increase indefinitely. This means that the sum or accumulated value of the series levels out to a specific number rather than growing without bound.
For the Laplace transform to exist for a function, the integral \( \int_0^\infty e^{-st} f(t) \, dt \) must converge.
This implies that as we take the integral from 0 to infinity, it approaches a finite number.
  • A convergent integral means that the area under the curve of \( e^{-st} f(t) \) becomes finite as \( t \rightarrow \infty \).
  • If the integral diverges, it means the total area is infinite, and the Laplace transform does not exist.
Hence, checking for convergence is often the first step when determining if a function has a Laplace transform.
Exponential Growth
Exponential growth is a rapid rate of increase that can cause functions to lack a Laplace transform. If a function grows exponentially, it means it increases at a rate proportional to its current value, leading to faster and faster growth over time.
Consider a function of the form \( f(t) = e^{at} \) where \( a > 0 \). This expresses a rapidly increasing exponential growth as time grows.
  • Such a function poses a problem when calculating the Laplace transform since it requires \( f(t) \) to grow more slowly than \( e^{st} \) as \( t \rightarrow \infty \).
  • When \( a \geq s \), the growth of \( f(t) = e^{at} \) outpaces the decay factor \( e^{-st} \), leading to an integral that does not converge.
As a result, understanding the concept of exponential growth is pivotal when considering the eligibility of a function for the Laplace transform.
Integral Divergence
Integral divergence occurs when an integral does not reach a finite value as its bounds stretch to infinity. Divergence indicates that as we attempt to add up the continuous inputs or area under a curve, the result keeps increasing indefinitely.
To check if a function lacks a Laplace transform, one can examine if the integral of the Laplace transformation diverges.
Take the function \( f(t) = e^{t^2} \). Here, the Laplace integral \( \int_0^\infty e^{-st} e^{t^2} \, dt \) can be examined.
  • The term \( e^{t^2-st} \) does not tend toward zero as \( t \rightarrow \infty \) for any finite \( s \), signifying persistent and growing values.
  • Thus, the integral diverges, leading to the absence of a Laplace transform for such functions.
Integral divergence highlights why certain fast-growing functions cannot be transformed using the Laplace integration.

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

Using differentiation, integration, s-shifting, or convolution (and showing the details), find \(f(t)\) if \(\mathscr{P}(f)\) equals: $$\frac{6}{(s+1)^{2}}$$

Using the Laplece transform and showing the details of your work, solve the initial value problem: $$y_{1}^{\prime}=y_{2}+1-u(t-1)$$$$y_{2}^{\prime}=-y_{1}+1-u(t-1), \quad y_{1}(0)=0$$$$y_{2}(0)=0$$

Find the Laplace transforms of the following functions. Show the details of your work. \((a, b, k, \omega, \theta\) are constants.) $$\cos (\omega t+\theta)$$

Showing the details, find, graph, and discuss the solution. $$\begin{aligned}&y^{\prime \prime}+2 y^{\prime}+5 y=25 t-1008(t-\pi)\\\&y(0)=-2, \quad y^{\prime}(0)=5\end{aligned}$$

Showing the details of your work, find \(\mathscr{L}(f)\) if \(f(t)\) equals: $$t \cos (t+k)$$
See all solutions

Recommended explanations on Physics Textbooks

Mechanics and Materials

Read Explanation

Atoms and Radioactivity

Read Explanation

Waves Physics

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Classical Mechanics

Read Explanation

Further Mechanics and Thermal Physics

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.