The Inverse Laplace Transform Calculator is a valuable mathematical tool used in engineering, physics, and various scientific fields to determine the original function in the time domain from its Laplace-transformed form in the frequency domain. It performs the reverse operation of the Laplace transform, enabling the retrieval of a function from its transformed representation.
Formula of Inverse Laplace Transform Calculator
The formula for the Inverse Laplace Transform is express as follows:
f(t) = L⁻¹{F(s)} = 1/(2πi) * ∫[c-i∞, c+i∞] e^(st) * F(s) ds
In this formula:
- f(t) represents the inverse Laplace transform of F(s) concerning time ‘t’.
- F(s) denotes the Laplace transform of the function intended to be inverted.
- s is the complex frequency variable.
- c is a real number greater than the real part of all singularities of F(s).
- The integral is taken over a contour in the complex plane, typically a vertical line to the right of all singularities of F(s).
- ‘i’ symbolizes the imaginary unit.
General Terms Table
| Term | Description |
|---|---|
| s | Complex frequency variable |
| t | Time |
| F(s) | Laplace transform of a function |
| f(t) | Inverse Laplace transform of F(s) |
| c | Real number greater than singularities |
This table provides a quick reference for commonly used terms related to the Inverse Laplace Transform.
Example of Inverse Laplace Transform Calculator
Suppose we have the Laplace-transformed function F(s) = 3s^2 + 2s + 5. Using the Inverse Laplace Transform Calculator, we aim to find the original function f(t).
Given the function F(s) = 3s^2 + 2s + 5, we apply the inverse Laplace transform formula to retrieve the function in the time domain, f(t). After calculations, we find that f(t) = 3t^2 + 2t + 5.
Most Common FAQs
The Inverse Laplace Transform is crucial in converting functions from the frequency domain to the time domain, allowing analysis and understanding of systems’ behavior.
The calculator facilitates engineers, scientists, and students in swiftly performing complex inverse transformations, enabling efficient problem-solving and analysis in various fields.