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The Laplace transform can be defined as an operator that is linear over a continuous function. It helps in mapping the domain of the function onto a complex plane thereby transforming the variables of the functions from the domain of time to the domain of frequency. This function’s inverse also occurs in the complex plane.
Inverse Laplace transform definition
To understand the inverse Laplace transform definition we first need to understand the definition of Laplace transform. A function is regarded as a piecewise function that is continuous if it has breaks that are finite in number and is not seen to be blowing up to infinity at any point. Let us suppose that the particular function f(u) is a continuous function, then f(u) can be defined through utilising the Laplace transform. Normally a function’s Laplace transform is denoted by L{f(u)} or F(s). The concept of Laplace transform is very important for solving different differential equations where the differential equation gets reduced into the form of an algebraic problem.
After this, the Inverse Laplace transform definition can be discussed which refers to the finding of the initial function from the Laplace transform function that is F(s). The inverse Laplace transform is normally given by the f(u) = L-1 {F(s)}.
For instance if we take the two Laplace transforms G(s) and F(s), then the inverse Laplace transform is given by L-1 {cG(s) + dF(s)} = c L-1 {G(s)} + d L-1, {F(s)}, Where c as well as d are constants. For this particular case, we can specifically take the different inverse transforms for different individual transforms, as well as their constant values in certain places to perform the appropriate operations and get the desired result.
Inverse Laplace transform formula
There are several Inverse Laplace transform formulas that help in calculating the inverse Laplace transform of a particular function. These have been outlined in the following.
| y (a) [Function] | Y (b) [Transform] | b |
| 1 | 1 / b | b is greater than 0 |
| a | 1 / b2 | b is greater than 0 |
| exp (ta), where t is a particular constant | 1 / (b-t) | b is greater than t |
| Ai where i is taken as an integer | i ! / s(I + 1) | b is greater than 0 |
| Sin (a), where s is a particular constant | t / (b2 + s2) | b is greater than 0 |
| cos (sa), where s is a particular constant | b / (b2 + s2) | b is greater than 0 |
| exp (ta) cos (sa) | s / (b – t)2 + s2 | b is greater than t |
| exp (ta) sin (ta) | (b – t) / (b – t)2 + s2 | b is greater than t |
Inverse Laplace transform calculator
Inverse Laplace transforms calculator is an online platform that helps in finding the inverse Laplace transform of a particular function easily. There are many online websites as well as platforms that offer this inverse Laplace transform calculator. To use this calculator one must have some prior knowledge of the Inverse Laplace transform. It should be remembered that L-1 [Y(c)] (d) is a particular function and the value of y(d) can be calculated via L (y(a) = Y(d). Hence, normally this linearity property is utilised to find the particular Inverse Laplace transform.
Voovers provides an efficient, free, online Inverse Laplace Transform Calculator. The entire page on this website has been written in JavaScript and has been powered by the CAS. In this calculator firstly the function which is entered is fed to the CAS. Major calculation occurs within the CAS where the functions are treated as a particular set of symbols. Here each character is preserved exactly as they were initially entered with zero truncation errors or round-off errors. It carries out the overall calculation in a similar manner a particular person would do with pencil and paper. The inverse Laplace calculator is very efficient and always yields correct results.
Conclusion
The overall article has been written on the core mathematical concept of Laplace transform and its inverse. An Agricultural Engineering student must know how to use the Inverse Laplace transform function properly. This core topic has been thoroughly analysed through sufficient discussion on inverse Laplace transform definition, inverse Laplace transforms formula, and inverse Laplace transform calculator.
