## Laplace Transform and Inverse Laplace Transform

Laplace Transform

The Laplace transform is a mathematical operation that takes a function of a real variable and transforms it into a function of a complex variable. The Laplace transform is used in many branches of mathematics, engineering, and physics.

Inverse Laplace Transform

The inverse Laplace transform is the operation that transforms a function of a complex variable back into a function of a real variable. The inverse Laplace transform is used to solve problems in many branches of mathematics, engineering, and physics.

## What is Laplace Transform:

The Laplace transform is a mathematical operation that allows us to calculate the derivative of a function using its integral. It is a function of a complex variable, and it is used to solve problems in physics and engineering.

## What is Inverse Laplace Transform:

The inverse Laplace transform is a mathematical function that takes a Laplace transform (s) of a function and returns the original function. It is usually denoted by the symbol L-1.

The inverse Laplace transform can be used to solve problems in physics and engineering. It is also used in the field of signal processing.

## Laplace Transform and Inverse Laplace Transform

Inverse Laplace Transform

If is a function of time, then the inverse Laplace transform is a function of .

The inverse Laplace transform is usually denoted by .

It is defined as:

where is the Laplace transform of .

Example

Find the inverse Laplace transform of

The inverse Laplace transform of is:

## Inverse Laplace Transform Table

Inverse Laplace Transform

This table is a collection of inverse Laplace transforms for common functions.

To use the table, find the function in the left column and then read across to the right to find the corresponding inverse Laplace transform.

If the function is not listed, then the inverse Laplace transform does not exist.

## Inverse Laplace Transform Theorems

Inverse Laplace Transform Theorem 1

If is a function and is its Laplace transform, then the inverse Laplace transform of is .

Inverse Laplace Transform Theorem 2

If is a function and is its Laplace transform, then the inverse Laplace transform of is .

Inverse Laplace Transform Theorem 3

If is a function and is its Laplace transform, then the inverse Laplace transform of is .

## Inverse Laplace Transform Examples

1. Find the inverse Laplace transform of

We will use the Laplace transform calculator to find the inverse Laplace transform of the given function.

The inverse Laplace transform of is .

2. Find the inverse Laplace transform of

We will use the Laplace transform calculator to find the inverse Laplace transform of the given function.

The inverse Laplace transform of is .

## :

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