Calculator
Choose a supported function family, enter parameters, and calculate the corresponding Laplace transform. The graph shows sampled transform magnitude along the positive real s-axis.
Example Data Table
Use these examples to verify notation and compare transform patterns before entering your own parameters.
| f(t) | Laplace Transform F(s) | Notes |
|---|---|---|
| 1 | 1 / s | Constant input |
| t^2 | 2 / s^3 | Power rule with n = 2 |
| e^(3t) | 1 / (s - 3) | Exponential shift |
| sin(4t) | 4 / (s^2 + 16) | Sine frequency example |
| e^(2t) cos(5t) | (s - 2) / ((s - 2)^2 + 25) | Shifted cosine |
| 3t^3 e^(t) | 18 / (s - 1)^4 | Scaled power with shift |
Formula Used
L{f(t)} = ∫ from 0 to ∞ of e^(-st) f(t) dt
L{A} = A / s
L{t^n} = n! / s^(n + 1)
L{e^(at)} = 1 / (s - a)
L{t^n e^(at)} = n! / (s - a)^(n + 1)
L{sin(bt)} = b / (s^2 + b^2)
L{cos(bt)} = s / (s^2 + b^2)
L{sinh(bt)} = b / (s^2 - b^2)
L{cosh(bt)} = s / (s^2 - b^2)
L{e^(at)sin(bt)} = b / ((s - a)^2 + b^2)
L{e^(at)cos(bt)} = (s - a) / ((s - a)^2 + b^2)
How to Use This Calculator
- Select the function family matching your time-domain expression.
- Enter amplitude A and any needed parameters n, a, or b.
- Set the graph range for the positive real s-axis.
- Press Find Laplace Transform to compute the result.
- Review the transform, formula used, and convergence condition.
- Inspect the graph to see how the sampled transform changes.
- Export the result or example table in CSV or PDF format.
FAQs
1) Can this calculator solve any symbolic expression?
It solves major transform families directly. These include constants, powers, exponentials, trigonometric functions, hyperbolic functions, and common shifted combinations. For arbitrary composite expressions, you may need algebraic decomposition first.
2) What does the region of convergence mean?
The region of convergence states where the Laplace integral converges. It tells you which real part of s makes the transform valid. This matters in control, circuits, and differential equations.
3) Why do I need the parameter a?
Parameter a shifts the transform horizontally in the s-domain. Positive a moves poles rightward. Negative a moves them leftward. It represents exponential growth or decay in the time-domain function.
4) Why do I need the parameter b?
Parameter b controls oscillation or hyperbolic growth. For sine and cosine, it sets angular frequency. For sinh and cosh, it changes pole locations and convergence requirements.
5) Why is n limited to nonnegative integers?
The implemented power formulas use n! and standard integer-power Laplace rules. Restricting n keeps the calculator reliable and clear for common educational and engineering cases.
6) What does the Plotly chart represent?
The chart shows sampled magnitude values of F(s) along the selected real-axis interval. It helps you visualize peaks, decay, and behavior near poles without doing manual substitutions.
7) What is included in the CSV and PDF downloads?
The result export includes your selected family, expression, transform, formula, region of convergence, parameters, and graph range. The example export includes the sample reference table.
8) Can I use this for checking homework steps?
Yes. Use the result, formula section, and examples to confirm pattern matching and parameter substitution. It is especially useful for verifying standard transforms before simplifying larger problems.