Laplace Transforms

Laplace TransformsExpected Educational ResultsLaplace TransformsExponential Order Activity 06Properties of Laplace TransformsDefintion: Exponential Order, Definition: Piecewise ContinuityTheorem: Existence of Laplace TransformTheorem: Linearity of the Laplace TransformInvestigation 07Investigation 08Check Your WorkUse Technology to Verify Laplace TransformCC BY-NC-SA 4.0

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Laplace Transforms

Definition: Laplace Transforms

Let be a function on . The Laplace Transform of is the function defined as

Exponential Order

Activity 06

Use the DESMOS graph below:

For each of the following , find , , and , if possible, such that , for all . Explain.

  1. .
  2. .
  3. .
  4. .
  5. .

Properties of Laplace Transforms

Defintion: Exponential Order,

A is said to be of exponential order if there exists constants and such that for all

Definition: Piecewise Continuity

A is said to be piecewise continuous on a finite interval if is continuous at every point in , except possibly for a finite number of points at which has a jump discontinuity. A is said to be piecewise continuous on if is piecewise continuous on for all .

Theorem: Existence of Laplace Transform

If is piecewise continuous on and of exponential order , then exists for .

Theorem: Linearity of the Laplace Transform

Let , , and be functions with exponential order alpha. Let be a constant. Then for .

Investigation 07

Assume the conditions are met, prove both statements of the Linearity of the Laplace transform theorem.

Investigation 08

Find the Laplace transform following functions:

  1. .
  2. .
  3. .
  4. .
  5. .

Check Your Work

Use Technology to Verify Laplace Transform

Mathematica

Warnings:

  1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: http://www.jjw3.com/TECH_Common_Functions.pdf.

  2. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

  3. The arguments for LaplaceTransform[ ] are:

    1. the function that the transform is being applied;
    2. the independent variable of the function;
    3. the independent variable for the transform.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Last Modified: Sunday, 8 November 2020 4:12 EDT