Calculator Input
Use t as the variable. Supported functions include sin, cos, tan, exp, log, ln, sqrt, abs, sinh, cosh, and powers with ^.
Example Data Table
| Piece | Interval | Function | Use case |
|---|---|---|---|
| 1 | [0, 1) | t | Rising ramp input |
| 2 | [1, 3) | 2 | Flat pulse level |
| 3 | [3, ∞) | exp(-t) | Decaying tail response |
Formula Used
For a stepwise function, the total transform is the sum of interval transforms. A finite piece uses ∫ab e−stf(t)dt. An open ended interval uses a numerical upper tail estimate. The unit step form is f(t)[u(t-a)-u(t-b)] for each finite segment.
How to Use This Calculator
- Enter a positive value for s.
- Add each interval start and end.
- Use inf for an interval that continues forever.
- Type a function of t for each active interval.
- Press calculate to see the transform above the form.
- Use CSV or PDF buttons to save the result.
Article: Stepwise Laplace Transforms
Why piecewise inputs matter
Many real signals do not follow one rule forever. A machine may start with a ramp. It may then hold a steady level. Later, the response may decay. A stepwise function describes that behavior with separate formulas on separate intervals. The Laplace transform converts this time based description into an s domain expression. This helps students, engineers, and analysts study systems more clearly.
How the transform is built
The calculator treats every interval as one small problem. It multiplies the selected formula by e raised to negative s times t. It then integrates across the interval. The final answer is the sum of all interval contributions. This method matches the definition of the Laplace transform. It is also easy to audit because every piece keeps its own contribution.
Working with unit steps
A piecewise function can also be written with unit step functions. A term starts at one boundary and stops at another. The form u(t-a)-u(t-b) creates a clean window. This makes the switching behavior visible. It also connects the calculation with the second shifting theorem. That theorem is useful when expressions are shifted after a delay.
Numerical accuracy
This page uses composite Simpson integration. The method samples each interval at many points. It gives strong accuracy for smooth functions. Very sharp jumps, large powers, or fast exponential growth may need careful interval choices. For infinite intervals, the page uses a practical tail estimate. A larger tail factor can improve slow decays, but it can also increase processing time.
Reading the outputs
The main result shows F(s) for your chosen s value. The table shows each contribution. The first graph displays the original stepwise function. The second graph shows how F(s) changes across a range of s values. Export buttons help save the numeric report for homework, notes, or classroom demonstrations. Always check that s is positive when using ordinary Laplace transforms.
FAQs
1. What is a stepwise function?
A stepwise function uses different formulas on different intervals. Each part applies only within its assigned start and end points.
2. What does the calculator return?
It returns an approximate Laplace transform value at your chosen s. It also shows each interval contribution and graph data.
3. Can I use infinity as an endpoint?
Yes. Enter inf or leave the end field blank. The calculator uses a practical tail approximation for that piece.
4. Which functions are supported?
You can use powers, constants, t, pi, e, sin, cos, tan, exp, log, ln, sqrt, abs, and hyperbolic functions.
5. Why must s be positive?
Positive s values help the exponential damping term control the integral. Many standard Laplace transforms require a convergence region.
6. Is the result symbolic?
The page gives a numerical transform with clear interval steps. It is best for practical checking, graphing, and comparison.
7. How should I enter powers?
Use the caret symbol. For example, enter t^2 for t squared, or exp(-t) for an exponential decay.
8. Why do infinite intervals use a tail factor?
A computer cannot integrate to infinity directly. The tail factor sets a far upper limit based on the selected s value.