Calculator Inputs
Formula Used
Common exact transforms used by this calculator:
| Signal f(t) | Laplace transform F(s) | Convergence condition |
|---|---|---|
| A | A / s | s > 0 |
| A·tn | A·n! / sn+1 | s > 0 |
| A·eat | A / (s - a) | s > a |
| A·sin(bt) | A·b / (s² + b²) | s > 0 |
| A·cos(bt) | A·s / (s² + b²) | s > 0 |
| A·sinh(bt) | A·b / (s² - b²) | s > |b| |
| A·cosh(bt) | A·s / (s² - b²) | s > |b| |
| A·eat·sin(bt) | A·b / ((s - a)² + b²) | s > a |
| A·eat·cos(bt) | A·(s - a) / ((s - a)² + b²) | s > a |
How to Use This Calculator
- Select the function family that matches your signal.
- Enter amplitude and any required parameters such as n, a, or b.
- Provide the transform value s where you want F(s).
- Choose a finite upper limit T for the numerical estimate.
- Set the number of integration steps for the trapezoidal method.
- Click the calculate button to display the result above the form.
- Review the exact transform, numerical estimate, error, and graph.
- Use the CSV or PDF buttons to export the result summary.
Example Data Table
| Function Family | Sample Input | Chosen s | Exact F(s) | Notes |
|---|---|---|---|---|
| Constant | A = 1 | 2 | 0.5 | Simple baseline transform. |
| Power Function | A = 1, n = 2 | 3 | 2 / 27 = 0.074074 | Uses n! / s^(n+1). |
| Exponential | A = 2, a = 1.5 | 4 | 0.8 | Converges because 4 > 1.5. |
| Sine | A = 1, b = 3 | 2 | 3 / 13 = 0.230769 | Oscillatory example. |
| Shifted Cosine | A = 1, a = 0.5, b = 2 | 3 | 0.243902 | Shifted transform with damping gap. |
FAQs
1. What does this calculator compute?
It evaluates the Laplace transform for several common time-domain functions. It shows the formal transform, checks convergence at your chosen s, and estimates the integral numerically over a finite interval.
2. Why are there exact and numerical results?
The exact result comes from a known transform formula. The numerical result approximates the integral directly. Comparing both helps verify accuracy and shows how finite truncation affects the estimate.
3. Why can the exact transform be marked undefined?
Some transforms only converge for certain s values. For example, A·e^(at) requires s > a. If your chosen s violates the convergence condition, the improper integral does not exist there.
4. What does the upper limit T control?
T is the cutoff for the finite numerical integral. Larger T can improve approximation to the infinite integral, but it may also increase runtime or cause overflow for rapidly growing signals.
5. How many integration steps should I use?
Use more steps for better numerical precision, especially for oscillatory functions. Around 1000 is a good starting point, while sharper or longer-range cases may need several thousand.
6. What does the graph show?
The chart plots F(s) over a valid s range using the exact formula. When your selected s converges, the graph highlights that specific point so you can compare it with nearby values.
7. Can I use negative parameters?
Yes. Negative amplitudes, rates, and frequencies are accepted. However, convergence and numerical stability still matter, so large growth rates or unsuitable s values may produce undefined or overflowed results.
8. What do the CSV and PDF exports include?
The exports capture the current summary shown on screen, including the signal, formal transform, convergence note, selected s, exact value, numerical estimate, and absolute error.