Calculator Inputs
Choose up to three intervals. Parameter meaning changes with the selected function type.
Example Data Table
This example matches the default setup and shows how a segmented function can be entered before transform sampling.
| Segment | Function Type | Definition | Interval |
|---|---|---|---|
| 1 | Constant | f(t) = 4 | 0 ≤ t ≤ 1 |
| 2 | Linear | f(t) = 2t - 1 | 1 ≤ t ≤ 3 |
| 3 | Exponential | f(t) = 3e^(-0.5t) | 3 ≤ t ≤ 5 |
Formula Used
For a piecewise function, the Laplace transform is evaluated interval by interval and then added together.
This calculator uses a trapezoidal numerical integration method on each active interval. The final transform value at each sampled s point is the sum of all segment contributions.
How to Use This Calculator
- Enter the starting, ending, and step values for s.
- Set the number of integration steps for numeric accuracy.
- Enable the segments you need.
- Select a function type for each active segment.
- Fill in coefficient, parameter, and interval values.
- Submit the form to view results above the calculator.
- Review the transform table and the Plotly graphs.
- Use the CSV and PDF buttons to export the output.
FAQs
1. What does this calculator compute?
It approximates the Laplace transform of a user-defined piecewise function. Each active segment is integrated over its own interval, then all interval contributions are added for every sampled s value.
2. Why are there multiple segments?
Piecewise functions often change formula across intervals. Multiple segments let you model switches in amplitude, trend, oscillation, or decay without rewriting the whole function as a single expression.
3. What does the parameter input mean?
Its meaning depends on the selected type. It becomes b for linear, n for power, k for exponential, and ω for sine or cosine. Constant functions do not need it.
4. Are the results exact?
The results are numerical approximations. Increasing integration steps generally improves stability and accuracy, especially for rapidly changing intervals or functions with oscillation and exponential growth or decay.
5. What happens if intervals overlap?
The transform table still sums each enabled interval contribution. For the original function graph, the first matching interval is shown. Non-overlapping intervals are recommended for a standard piecewise definition.
6. Can I use negative interval values?
Yes, but interpret the model carefully. Classical Laplace problems usually start at nonnegative time. Negative inputs are still sampled numerically, though some power expressions may be limited for fractional exponents.
7. What does the Plotly graph show?
One graph samples the original piecewise function across its active interval range. The second graph shows approximate Laplace transform values F(s) across the s range you entered.
8. What is included in the exports?
The CSV export contains the sampled transform table. The PDF export includes summary values, active segment formulas, and the transform table for quick reporting or archiving.