Calculator Inputs
Use the general settings first. Then enable the transform families you want to combine. The page remains single column, while the calculator area uses the requested responsive grid.
Formula used
Linearity: L^-1{F(s) + G(s)} = f(t) + g(t). The calculator evaluates each enabled term separately, then adds the time-domain responses.
First-order poles: L^-1{A / (s + a)} = A e^(-at) and L^-1{A / (s - a)} = A e^(at).
Repeated poles: L^-1{A / (s + a)^n} = A t^(n-1)e^(-at)/(n-1)! and L^-1{A / (s - a)^n} = A t^(n-1)e^(at)/(n-1)!.
Cosine terms: L^-1{A(s + a)/((s + a)^2 + b^2)} = A e^(-at)cos(bt). The growth version replaces (s + a) with (s - a) and uses e^(at).
Sine terms: L^-1{Ab/((s + a)^2 + b^2)} = A e^(-at)sin(bt). The growth version uses (s - a) and e^(at).
How to use this calculator
- Set the numerical window with start time, end time, and step size.
- Enable one or more transform families from the six available term cards.
- Enter coefficient
A, shifta, and frequencybor powernwhen needed. - Submit the form to build the full s-domain expression and its inverse transform.
- Review the final expression, term-by-term mapping, summary values, graph, and numerical table.
- Use the CSV and PDF buttons to export the generated dataset and report.
Example data table
This sample reference table helps students connect standard s-domain forms with familiar time-domain functions.
| Example | F(s) | f(t) | Notes |
|---|---|---|---|
| 1 | 5 / (s + 2) |
5e^(-2t) |
Simple decaying exponential. |
| 2 | 3 / (s - 1) |
3e^(t) |
Simple growing exponential. |
| 3 | 4 / (s + 1)^3 |
2t^2e^(-t) |
Repeated pole with factorial scaling. |
| 4 | 2(s + 1) / ((s + 1)^2 + 9) |
2e^(-t)cos(3t) |
Shifted cosine response. |
| 5 | 12 / ((s + 1)^2 + 9) |
4e^(-t)sin(3t) |
Shifted sine response, since numerator equals Ab. |
FAQs
1. What expressions can this calculator invert?
It handles common inverse Laplace families: first-order poles, repeated poles, shifted sine terms, and shifted cosine terms. Multiple enabled terms are combined through linearity.
2. Does it parse arbitrary symbolic input?
No. It uses a structured term builder. That approach avoids parsing errors and keeps the results consistent for classroom and practical engineering-style calculations.
3. Why does the sine family ask for frequency b?
For sine responses, the numerator in the s-domain is proportional to b. The calculator uses your amplitude and frequency to construct that numerator correctly.
4. What does the shift a control?
The shift moves poles left or right in the s-domain. In time, it becomes exponential decay for s + a and exponential growth for s - a.
5. Why is there a power input n?
Repeated poles create polynomial factors in time. For example, 1/(s+a)^3 becomes a term involving t^2e^(-at) divided by 2!.
6. What does the graph show?
The graph plots the reconstructed time-domain signal over your chosen interval. It helps you inspect decay, growth, oscillation, and sign changes visually.
7. What is included in the CSV and PDF exports?
Both exports include the generated numerical dataset. The PDF also includes the main expressions, summary metrics, and a compact report layout for sharing or printing.
8. Can I use negative values for a or b?
The calculator treats shift and frequency as magnitudes. Their physical meaning is controlled by the chosen family, such as decay versus growth.