The inverse Laplace transform calculator is a digital tool that works backward to reveal the original mathematical expressions (real variables) from a transformed version called Laplace Transform. This tool aids in understanding the behavior of systems described in the Laplace domain.
The inverse Laplace transform is a method in which we can revert back from the frequency domain F(s) to the time domain f(t).
Inverse Laplace Transforms are, as its name suggests, the inverse operation of Laplace Transforms.
The Laplace Transform helps you change a function from a time domain f(t) to a frequency domain F(s). On the other hand, the inverse Laplace Transform reverses this transformation.
Inverse Laplace transform is also identified as inverse Laplace transformation or inverse Laplace and expressed as $$ L^{-1}f(t) $$.
Here's the formula to calculate the inverse Laplace transform,
$$ f(t) = \mathcal{L}^{-1}F(s) $$
In this formula:
$$ {f}{(t)} $$ represents the original function in the time domain.
$$ L^{-1} $$ denotes the inverse Laplace transform.
$$ {F}{(s)} $$ stands for the transformed function in the Laplace domain.
Here's a table of common examples of inverse Laplace transforms,
Laplace Transform (F(s)) | Inverse Laplace Transform (f(t)) |
---|---|
1 | δ(t) (Dirac delta function) |
e^(at) | 1/(s-a) |
t^n (for n ≥ 0) | n!/(s^(n+1)) |
e^(at)sin(bt) | b/((s-a)^2 + b^2) |
e^(at)cos(bt) | (s-a)/((s-a)^2 + b^2) |
1/(s-a) | e^(at) |
1/(s^2) | t |
1/(s(s-a)) | 1 - e^(at) |
We are going to solve an example through which you can easily calculate the inverse Laplace transform.
Let's find the inverse Laplace transform:
$$ F(s) = \frac{12}{s} - \frac{1}{s - 10} + 8(s - 25) $$
Upon examination of the terms, we notice that the denominator of the first term is simply a constant. The correct numerator for this term is "1". When using the inverse Laplace Transform Calculator with steps, we will only consider the factor 12 before performing the inverse transformation. Therefore, a = 10 is the correct numerator, just as it needs to be.
The third term appears to be exponential, but this time, a = 25. Before performing the inverse transformation, we need to factor in the 8.
In more detailed form:
$$ F(s) = \frac{12}{s} - \frac{1}{s - 10} + 8(s - 25) f(t) = 12(1) - e^{10t} + 8e^{25t} = 12 - e^{10t} + 8e^{25t} $$
So, the inverse transform of $$ {F}{(s)} is {f}{(t)} = 12 - e^{10t} + 8e^{25t} $$. You can also try the inverse Laplace calculator for hassle-free and faster results.
If you don’t want to solve manually then you can use this free online inverse laplace transform calculator for this. All you need to do is follow simple steps which include:
What to do:
What you get:
Here are some properties that describe the inverse Laplace of a function.
There are two types of Laplace transformation such as,
Libretexts.org: The Inverse Laplace Transform and Definition of the Inverse Laplace Transform
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