Inverse Laplace Calculator

Inverse Laplace Transform:

\[ f(t) = \frac{1}{2\pi i} \int F(s) e^{s t} ds \]

f(t) (Function):

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1. What is the Inverse Laplace Transform?

The Inverse Laplace Transform converts a function from the complex frequency domain (s-domain) back to the time domain. It is defined by the Bromwich integral:

\[ f(t) = \frac{1}{2\pi i} \int_{\gamma - i\infty}^{\gamma + i\infty} F(s) e^{s t} ds \]

2. How Does the Calculator Work?

The calculator uses the inverse Laplace transform formula:

\[ f(t) = \frac{1}{2\pi i} \int F(s) e^{s t} ds \]

Where:

\( F(s) \) — Laplace transform of the function
\( s \) — Complex frequency variable
\( t \) — Time variable
\( f(t) \) — Original function in time domain

Explanation: The integral is evaluated along a vertical line in the complex plane where the real part is greater than all singularities of F(s).

3. Importance of Inverse Laplace Transform

Details: The inverse Laplace transform is crucial for solving differential equations, analyzing control systems, and understanding the time-domain behavior of systems described in the frequency domain.

4. Using the Calculator

Tips: Enter the Laplace transform F(s), the complex variable s, and the time value t. The calculator will compute the corresponding time-domain function f(t).

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can be transformed?
A: The inverse Laplace transform works on functions that are Laplace transformable, typically piecewise continuous functions of exponential order.

Q2: How is the Bromwich integral evaluated?
A: The integral is typically evaluated using residue theorem, partial fractions, or numerical methods for complex cases.

Q3: What are common applications?
A: Electrical circuit analysis, mechanical systems, control theory, signal processing, and solving differential equations.

Q4: Are there limitations to this transform?
A: The transform requires that F(s) converges in some region of the complex plane and may not exist for functions that grow too rapidly.

Q5: How does this relate to Fourier transform?
A: The Laplace transform is a generalization of the Fourier transform that can handle a broader class of functions and includes exponential weighting.