Inverse Laplace Transform Examples

Inverse Laplace transform 

The inverse Laplace transform calculator is used in different fields of math to convert the function of complex variables (usually time) into one with real variables (usually frequency). Get the complete calculation including the explanation of each step. 

How to use this tool? 

There is basically one step to using this inverse Laplace calculator and that is entering the function. All you have to do is Input your problem question and click calculate

What is the inverse Laplace transform?

The Inverse Laplace Transform, as the name suggests, is the inverse operation of the Laplace Transform. It enables us to revert from the frequency domain back into the time domain.

Where the Laplace Transform aids in transforming a function from the time domain (function of time 't') into the frequency domain (function of complex number 's'), the Inverse Laplace Transform reverses this process.


The formula for the Inverse Laplace Transform is given by:

f(t) = L^-1{F(s)} = (1 / 2πi) ∫_γ-i∞^γ+i∞ e^st F(s) ds

This formula represents a complex contour integral, where 'γ' is a real number so that the contour path passes through the real part of 's' = γ.

In essence, the Inverse Laplace Transform facilitates the solution of linear ordinary differential equations and systems of such equations.

How to calculate the inverse Laplace?

Let's consider an example to further clarify the process of calculating an Inverse Laplace Transform. Please note, this does require a basic understanding of integral calculus, complex numbers, and contour integration.

Let's find the inverse Laplace Transform of F(s) = (s + 3) / (s^2 + 2s + 5).

The first step is to complete the square in the denominator:

F(s) = (s + 3) / (s^2 + 2s + 4 + 1) = (s + 3) / ((s + 1)^2 + 1).

At this point, manipulate the function to match a form from the table of Laplace Transforms. One common entry is: 

L{e^at cos(bt)} = s / (s^2 + 2as + a^2 + b^2) and L{e^at sin(bt)} = b / (s^2 + 2as + a^2 + b^2). 

Notice the denominator in these entries matches the denominator of our function.

To utilize these forms, we rewrite F(s) = (s + 3) / ((s + 1)^2 + 1) as:

F(s) = [s + 1 + 2] / ((s + 1)^2 + 1) = (s + 1) / ((s + 1)^2 + 1) + 2 / ((s + 1)^2 + 1).

Now, each part of the right-hand side of this equation matches one of the Laplace Transforms mentioned earlier, and we find:

L^-1{(s + 1) / ((s + 1)^2 + 1)} = e^-t cos(t) and L^-1{2 / ((s + 1)^2 + 1)} = 2e^-t sin(t).

Thus, the inverse Laplace Transform of the original function is:

f(t) = e^-t cos(t) + 2e^-t sin(t).

For more complex problems, you may need to use partial fraction decomposition, where you express the function as a sum of simpler fractions. 

Laplace transform table

Here's a basic table of some common Inverse Laplace Transforms. It lists the Laplace Transform in the frequency domain F(s) and its corresponding time domain function f(t):









s/(s^2 + a^2)


a/(s^2 + a^2)






sF(s) - f(0)



∫_0^t f(u) du


u_c(t) (unit step function)


δ(t) (Dirac delta function)

In this table:

  • n is a positive integer
  • a is a real constant
  • f(t) is a given function, and F(s) is its Laplace transform
  • u_c(t) is the unit step function which is 0 for t < c and 1 for t >= c
  • δ(t) is the Dirac delta function, which is a function that is zero everywhere except 0, and has an integral over the entire real line equal to 1

Uses and applications:

The Inverse Laplace Transform plays a vital role in a variety of fields by offering a means to switch between the time and frequency domains. This ability can simplify the process of solving linear differential equations, particularly in the context of initial value problems.

 Here are a few applications:

  • Control Theory and Engineering: The Inverse Laplace Transform is extensively used in analyzing and controlling dynamic systems. For example, engineers use it to understand the time-domain behavior of electronic circuits and control systems from their transfer functions in the s-domain.
  • Signal and Image Processing: In signal and image processing, the Inverse Laplace Transform can be used to convert frequency-domain representations of signals back into the time-domain, which can be easier to analyze and interpret.
  • Physics: In the study of various physical phenomena involving heat conduction, vibrations, and electric circuits, mathematical models often end up as linear differential equations. The Laplace Transform, and subsequently the Inverse Laplace Transform, is a key tool in solving these equations.

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