Calculator form
Enter a supported function family, then compare your proposed transform using algebraic form with s, numbers, parentheses, and + - * / ^.
Example data table
| Case | Function family | Input f(t) | Expected F(s) | ROC |
|---|---|---|---|---|
| 1 | Constant | 5 |
5/s |
Re(s) > 0 |
| 2 | Polynomial | 2*t^3 |
12/s^4 |
Re(s) > 0 |
| 3 | Exponential | 4*e^(2t) |
4/(s-2) |
Re(s) > 2 |
| 4 | Sine | 3*sin(2t) |
6/(s^2+4) |
Re(s) > 0 |
| 5 | Shifted cosine | 2*e^(3t)*cos(4t) |
2*(s-3)/(((s-3)^2)+16) |
Re(s) > 3 |
Formula used
This checker uses standard Laplace transform pairs and the exponential shift rule. The calculator compares your algebraic proposal with the computed reference form.
| Supported family | Transform rule |
|---|---|
| Constant | L{A} = A/s |
| Polynomial | L{A*t^n} = A*n!/s^(n+1) |
| Exponential | L{A*e^(at)} = A/(s-a) |
| Sine | L{A*sin(bt)} = A*b/(s^2+b^2) |
| Cosine | L{A*cos(bt)} = A*s/(s^2+b^2) |
| Hyperbolic sine | L{A*sinh(bt)} = A*b/(s^2-b^2) |
| Hyperbolic cosine | L{A*cosh(bt)} = A*s/(s^2-b^2) |
| Shifted sine | L{A*e^(at)*sin(bt)} = A*b/((s-a)^2+b^2) |
| Shifted cosine | L{A*e^(at)*cos(bt)} = A*(s-a)/((s-a)^2+b^2) |
How to use this calculator
- Choose the function family that matches your time-domain expression.
- Enter amplitude A, then complete n, a, or b when that family needs them.
- Type your proposed Laplace transform in algebraic form using s, numbers, parentheses, and basic operators.
- Press Check Transform to display the result above the form.
- Review the expected transform, convergence condition, steps, and sample value comparison table.
- Use the CSV button for history export or the PDF button for a clean study copy.
FAQs
1. What does this checker verify?
It computes the reference Laplace transform for supported function families and compares your typed proposal against that result using safe numerical test values of s.
2. Can I type any symbolic expression?
Use only numbers, the variable s, parentheses, and the operators +, -, *, /, and ^. Text functions or unsupported notation will stop the parser.
3. Why is the region of convergence shown?
The region of convergence tells you where the transform exists. It also helps the checker choose safe sample values of s for equivalence testing.
4. Does an equivalent algebraic form still pass?
Yes. Equivalent algebraic forms can pass even when the text does not exactly match, because the checker compares evaluated values at multiple safe points.
5. Why does the polynomial option need an integer n?
This version uses the standard power rule with factorials. That rule is defined here for non-negative integers, which keeps the workflow clear and reliable.
6. Can I export the results?
Yes. Use the CSV button to download recent calculation history and the PDF button to save the visible result area as a printable document.
7. Which transform families are supported?
The calculator supports constants, powers, exponentials, sine, cosine, hyperbolic sine, hyperbolic cosine, shifted sine, and shifted cosine forms.
8. Is this a full symbolic algebra system?
No. It is a structured checker for common instructional forms. It is excellent for coursework, practice, and quick verification of standard transform pairs.