Inverse Laplace Transform Calculator

Inverse Laplace Transform calculator will convert the complex function F(s) into a simple function f(t) in the real-time domain.

Our calculator has several properties that make it useful for analyzing linear dynamical systems. So, start learning to understand how to the inverse Laplace transform of the functions with the help of an example and inverse Laplace table.

What is Inverse Laplace Transform?

In mathematics, the inverse Laplace transform online is the opposite method, starting from F(s) of the complex variable s, and then returning it to the real variable function f(t). Ideally, we want to simplify the F(s) of the complex variable to the point where we make the comparison for the formula from an inverse Laplace transform table.

However, an Online Laplace Transform Calculator provides the transformation of the real variable function to the complex variable.

Inverse Laplace Transform Formula:

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

$$ L {f}(s) = L {f(t)} (s) = F(s) $$

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

If given the two Laplace transforms G (s) and F (s), then

$$L^{−1} {xF(s) + y G(s)} = x L^{−1} {F(s)} + y L^−1{G(s)} $$

With any constants x and y.

How to Find Inverse Laplace Transform?

There are many inverse Laplace transform online examples available for determining the inverse transform.

Example 1:

Find the inverse transform:

$$F(s) = 21 / s − 1 /(s − 17) + 15 (s − 33) $$

Solution:

As can be seen from the denominator of the first term, it is just a constant. The correct numerator of this term is "1". If we use the inverse Laplace Transform Calculator with steps free, then we will only consider factor 21 before the inverse transformation. Therefore, a = 17 is a numerator which exactly what it needs to be. The third term also seems to be exponential, but this time a = 33, we need to factor the 15 before performing the inverse transformation.

More details than what we usually enter,

$$ F(s) = 21/s − 1 / (s − 17) + 15 (s − 33) $$

$$ f(t) = 21(1) − e^{17t} + 15 (e^{33t}) $$

$$ = 21 − e^{17t} + 15 e^{33t} $$

How Inverse Laplace Transform Calculator Works?

An online  inverse Laplace calculator with solution allows you to transform a complex Function F(s) into a simple real function f(t) by following these instructions:

Input:

• Enter a complex function F(s) and see the equation preview in Laplace form.
• Hit the calculate button to see the results.

Output:

• The Laplace inverse calculator with steps transforms the given equation into a simple form.
• You can transform many equations with this Laplace step function calculator numerous times quickly without any cost.

Reference: 

From the source of Wikipedia: Inverse Laplace transform, Mellin's inverse formula, Post's inversion formula.

From the source of Paul's online notes: Inverse Laplace Transform, Factor in the denominator, Term in partial fraction decomposition.