Solver inputs
Use up to four additive terms. Zero coefficients are ignored automatically. The layout stays single-column overall, while form fields adapt to screen size.
Supported function families
This structured solver supports linear combinations of common Laplace-transform families and handles each term symbolically before building a combined numeric response.
Formula used
The core definition is:
L{f(t)} = ∫0∞ e-st f(t) dt
For practical solving, the calculator applies standard transform identities term-by-term and then adds the resulting transforms because the Laplace operator is linear.
| Time-domain term | Transform rule | Key condition |
|---|---|---|
| 1 | 1/s | Re(s) > 0 |
| tn | n!/sn+1 | n is a nonnegative integer |
| eat | 1/(s-a) | Re(s) > a |
| sin(bt) | b/(s2+b2) | Re(s) > 0 |
| cos(bt) | s/(s2+b2) | Re(s) > 0 |
| u(t-a)g(t-a) | e-asG(s) | Second shifting theorem |
How to use this calculator
- Choose a function family for each active term.
- Enter the coefficient and the needed parameters only.
- Set unused terms to zero if you need fewer components.
- Adjust time and s ranges to control the Plotly graphs.
- Press the solve button to place the result below the header.
- Review the transform, poles, ROC, and sample tables.
- Use the CSV and PDF buttons to save the output.
Example data table
| Example input f(t) | Expected Laplace transform | ROC |
|---|---|---|
| 3 | 3/s | Re(s) > 0 |
| 2t2 | 4/s3 | Re(s) > 0 |
| 5e-2t | 5/(s+2) | Re(s) > -2 |
| 4sin(3t) | 12/(s2+9) | Re(s) > 0 |
| u(t-2)cos(3(t-2)) | e-2ss/(s2+9) | Re(s) > 0 |
FAQs
1. What does this solver handle best?
It handles structured Laplace problems built from common transform families. You can combine up to four terms, inspect each symbolic rule, and see the summed transform instantly.
2. Can it parse any typed expression automatically?
This version uses guided term selection instead of unrestricted expression parsing. That keeps the mathematics reliable and makes the transform steps, poles, and convergence checks easier to verify.
3. Why does the calculator show a region of convergence?
The Laplace transform exists only where the defining integral converges. The ROC tells you the valid real-part boundary for s after all active terms are combined.
4. What do the poles represent?
Poles are singular points where the transform becomes unbounded or undefined. They are useful in systems analysis, stability work, transfer functions, and inverse-transform reasoning.
5. Why are some sampled F(s) values undefined?
That happens when the chosen real s value lands on a pole or creates division by zero. The calculator marks those points as undefined rather than showing misleading numbers.
6. How are delayed terms transformed?
Delayed terms use the second shifting theorem. A delay a introduces the exponential factor e-as in the transform while preserving the base transform structure.
7. What does the Plotly graph show?
One graph shows the combined time-domain function over your chosen t range. The other shows the computed transform along the selected positive real s axis.
8. Can I export the results?
Yes. The page includes CSV export for summary and sampled data, plus PDF export for the summary and the visible result tables.