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# Inverse Laplace Transform Calculator – Variable Transformation

A real-time inverse Laplace transform calculator will convert the complex function F(s) into a simple function f(t). Several properties of the inverse Laplace calculator make it useful for analyzing linear dynamical systems. You can now learn how to find the inverse Laplace transform using an example and inverse Laplace table.

## Introduction of Inverse Laplace Transform:

As in mathematics, the inverse Laplace transform starts with F(s) of the complex variable s and returns that to the real variable function f(t). Ideally, we want to simplify the F(s) of the complex variable to the point where we can compare it with the formula from an inverse Laplace transform table.

If the Laplace inverse formula for each part of F(s) can be found on the table, then the inverse Laplace Transform calculator can be used. Let us begin from the perspective of the real variable t and obtain the function of f(t).

With an Online Laplace Transform Calculator, you can transform a variable function into a complex variable.

### Laplace Inverse Transform Formula:

A real function f(t), which is piecewise continuous and exponentially restricted, is the inverse Laplace transform of a function F(s). Its properties include:

Lf(s) = Lf(t)(s) = F(s)

It can be demonstrated that if the function F(s) has the inverse Laplace transform f(t), then F(t) is uniquely determined (considering the function is divided only by a set of distinguishing points, use Null Lebesgue metric).

Given two Laplace transformations G (s) and F (s), then

L-1 xF(s) + yG (s) = xL-1 F(s) + yL1G(s)

With any constants x and y.

### Inverse Laplace Transform – How to Find?

For determining the inverse Laplace transform, there are many examples available.

• Example No 1:

Find the inverse transform:

F(s) = 21/s – 1/(s-17) + 15(s-33)

Solution:

In this case it is a constant, as can be seen from the denominator of the first term. The denominator of this term should be “1”. The inverse Laplace Transform Calculator only considers factor 21 before the inverse transformation. Therefore, a = 17 is a numerator that is exactly what it needs to be. It appears that the third term is also exponential, but this time a = 33, which means we need to factor 15 before performing the inverse transformation.

f(s) = 21/s – 1/(s-17) + 15(s-33)

f(t) = 21(1) – e17t + 15 (e33t)

= 21 – e17t + 15e33t

• Example No 2:

Determine the inverse Laplace transform:

f(s) = 9s/ (s2 +36) +3/ (s2 + 36)

Solution:

Numerators in an inverse Laplace table tell us what denominator we really have. Here, the number 9 is multiplied by the numerator. Only one constant appears in the numerator of the second term. Therefore, the value of inverse Laplace must be multiplied by six:

F(s) = 9s/ (s2 + (6)2) + 7(6/6)/s2 + (6)2

= 9s/ (s2 + (6)2) + (7/6) 6/s2 + (6)2

By taking the inverse transform,

f(t) = 9cos (6t) + 7/6sin (6t)

Alternatively, you can obtain the same results by substituting these values in the inverse Laplace Transform Calculator.

## How does an Inverse Laplace Transform Calculator work?

Following these instructions, you can transform a complex function F(s) into a simple real function f(t) using an online inverse Laplace calculator:

• Input:
1. You can preview the equation in Laplace form by entering a complex function F(s).
2. Click the calculate button to see the results.
• Output:
1. Calculate the inverse of a given equation using the Laplace inverse calculator.
2. This Laplace step function calculator allows you to transform many equations numerous times without incurring any costs.

## How do you determine the inverse of Laplace’s constant?

The sum of the two terms is the reciprocal of the sum of the inverse transformations of the two terms, with the latter being taken into account separately. Basically, a Laplace transform is a constant multiplied by a function with an inverse constant.

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