Inverse Laplace TransformsExpected Educational ResultsInverse Laplace TransformsDefinition: Inverse Laplace TransformIntegral Form for Inverse Laplace TransformLinearity of the Inverse Laplace TransformTheorem: Linearity of the Inverse Laplace TransformUse Technology to use the method of partial fractionsCheck Your WorkUse Technology to Verify Inverse Laplace TransformInvestigation 01CC BY-NC-SA 4.0

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

**Section 7.4**– Inverse Laplace Transforms

**Objective 21–1**: I understand the Laplace transform as a map of a function onto a specific class of functions.**Objective 21–2**: I can find the inverse Laplace Transform of a given transform.

Given , if there is a function that is continuous on and satisfies then is the **inverse Laplace transform** of , then

Here is the formula for calculating the inverse Laplace transform:

where the integral is over a line in the complex plane for suitable .

**NOTE**: It will be easier to use the method of partial fractions and your knowledge of Laplace transforms to compute an inverse Laplace transform!

Assume , , and exist and are continuous on . Let be a constant. Then

and

.

**NOTE**: Often the Method of Partial Fractions will be needed to compute an inverse Laplace transform!

**Example 01**:

Find the inverse Laplace transform of .

**Solution**:

Using method of partial fractions:

**Mathematica**

`1``(* Separate into partial fractions: 1/(s^2-3s-10) *)`

2`Apart[1/(s^2-3s-10)]`

**Warnings**:

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

Using the linearity property of inverse Laplace transforms:

Using our knowledge of Laplace transforms:

**Mathematica**

`xxxxxxxxxx`

21`(* Example from Example 01: Find inverse Laplace transform of 1/(s^2-3s-10) *)`

2`InverseLaplaceTransform[1/(s^2-3s-10), s, t]`

**Warnings**:

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

The arguments for

*InverseLaplaceTransform[ ]*are:- the function that the inverse transform is being applied;
- the independent variable of the function;
- the independent variable for the inverse transform.

**Example 02**:

Find the inverse Laplace transform of .

**Solution**:

Rewrite using algebra:

Use the linearity property of inverse Laplace transforms:

Rewriting using algebra:

Using our knowledge of Laplace transforms:

**Example 03**:

Find the inverse Laplace transform of .

**Solution**:

Rewrite using algebra:

Rewrite using algebra:

Using the linearity property of inverse Laplace transform:

Using our knowledge of Laplace transforms:

Determine the inverse Laplace transforms of the given functions:

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**Last Modified**: Sunday, 8 November 2020 17:10 EDT