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Inverse Laplace Transform Calculator

Enter a function in the frequency domain, and the calculator will transform it into its corresponding time-domain function using the inverse Fourier transform.

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Inverse Laplace Transform Calculator

The Inverse Laplace Transform Calculator converts a given complex function F(s) into its corresponding real-time function f(t). This tool is especially useful for analyzing linear dynamical systems and solving differential equations efficiently.

Our calculator also demonstrates the process using examples and reference tables, making verification quick and intuitive.

What is the Inverse Laplace Transform?

In mathematics, the inverse Laplace transform is the process of converting a function F(s) in the complex s-domain back into its original time-domain function f(t). Essentially, it reverses the Laplace transformation to recover f(t). Using an inverse Laplace transform table simplifies comparison and solution.

Similarly, an Online Laplace Transform Calculator converts time-domain functions into the s-domain.

Inverse Laplace Transform Formula:

The inverse Laplace transform converts F(s) into a piecewise continuous and exponentially bounded function f(t). Key properties include:

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

If F(s) has an inverse Laplace transform f(t), it is uniquely determined (except on sets of measure zero).

Linearity property for two Laplace transforms F(s) and G(s):

$$ L^{-1} \{ x F(s) + y G(s) \} = x L^{-1} \{ F(s) \} + y L^{-1} \{ G(s) \} $$

where x and y are constants.

How to Find the Inverse Laplace Transform?

Several online examples guide the step-by-step process of finding inverse Laplace transforms.

Example 1:

Find the inverse transform of:

$$ F(s) = \frac{21}{s} - \frac{1}{s - 17} + 15 (s - 33) $$

Solution:

First term: $$ \frac{21}{s} $$ corresponds to a constant 21 in the time domain.
Second term: $$ \frac{-1}{s-17} $$ corresponds to an exponential $$ -e^{17t} $$ (shift property with a = 17).
Third term: $$ 15(s-33) $$ corresponds to $$ 15 e^{33t} $$ (multiplied by 15 and shifted by a = 33).

Combining all terms:

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

An inverse Laplace Transform Calculator with steps simplifies this process significantly.

How the Inverse Laplace Transform Calculator Works

Input:

Enter the complex function F(s) and preview the Laplace equation.
Click the Calculate button to process the inverse transformation.

Output:

The calculator provides the simplified f(t) function with step-by-step explanation.
Supports multiple equations and repeated transformations efficiently and free of cost.

References:

Wikipedia: Inverse Laplace Transform – includes Mellin’s inversion formula and Post’s inversion formula.
Paul's Online Notes: Inverse Laplace Transform – covers partial fraction decomposition and denominator factors.

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