Analyze standard signals, shifted functions, damping, and poles with fast calculations now. Build insight daily. Visualize transforms, compare responses, and export clean results easily.
| # | Signal | Laplace transform | Region of convergence |
|---|---|---|---|
| 1 | u(t) |
1 / s |
Re(s) > 0 |
| 2 | e^(2t)u(t) |
1 / (s - 2) |
Re(s) > 2 |
| 3 | t^2u(t) |
2 / s^3 |
Re(s) > 0 |
| 4 | sin(4t)u(t) |
4 / (s^2 + 16) |
Re(s) > 0 |
| 5 | cos(3t)u(t) |
s / (s^2 + 9) |
Re(s) > 0 |
| 6 | e^(-t)sin(5t)u(t) |
5 / ((s + 1)^2 + 25) |
Re(s) > -1 |
| 7 | (t - 2)u(t - 2) |
e^(-2s) / s^2 |
Re(s) > 0 |
| 8 | sinh(2t)u(t) |
2 / (s^2 - 4) |
Re(s) > 0 |
The unilateral Laplace transform is defined by L{f(t)} = ∫0∞ e-stf(t)dt. The calculator uses closed form identities for common causal signals. Those identities help you avoid repeated manual integration.
For powers, the transform is A·n! / sn+1. For exponentials, it becomes A / (s - a). For sine and cosine, the denominator depends on s2 + b2. Shifted signals use the time shift rule and introduce the factor e-cs.
Poles come from denominator zeros. The region of convergence depends on growth and causality. Graphs show both the time signal and the real axis transform behavior across the chosen s range.
A graphing Laplace transform calculator helps you connect formulas with behavior. That matters in maths, physics, and engineering. Many students memorize transform pairs but never see how the original signal changes the transform curve. This tool closes that gap.
You can test exponentials, powers, sine waves, cosine waves, shifted signals, and damped oscillations. Each choice creates a direct mapping from time domain to s domain. You can inspect poles, check the region of convergence, and compare sample values. Those steps build intuition fast.
Laplace transforms simplify differential equations. They also support transfer function analysis, stability checks, and transient studies. When you graph the source signal, you see amplitude, oscillation, delay, and damping. When you graph the transform, you see how growth and frequency affect the denominator and the pole pattern.
The calculator is useful for unilateral transform work because many textbook problems assume causal signals. A causal signal begins at or after zero. That assumption changes the region of convergence and makes the time shift rule important. When you enter a delayed power term, the output preserves the pole order and adds an exponential shift factor. Seeing that result beside the graph makes the theorem easier to remember.
For example, a pure exponential shifts the pole location. A sine term creates conjugate poles on the imaginary axis. A delayed signal adds an exponential factor in the transform. That factor changes phase related behavior and preserves the underlying pole order. A hyperbolic signal can create real poles and a very different graph shape. These comparisons are hard to spot from a formula list alone.
Tables are useful, but graphs make trends visible. You can spot rapid decay, rising magnitude, and undefined values near poles. That helps when checking homework, preparing lessons, or validating control system expressions. The result section also keeps export tools nearby, so you can save clean data for reports, class notes, or revision sheets.
This calculator is practical because it combines formulas, graphs, examples, and export options on one page. You get a compact workflow, faster checking, and clearer understanding. It also helps you verify manual integration steps before moving into inverse transforms, transfer functions, and response problems.
It computes closed form Laplace transforms for supported causal signal families. It also shows poles, the region of convergence, time samples, transform samples, and two graphs.
No. It focuses on structured signal families with reliable formulas. That keeps calculations fast, readable, and practical for many teaching and analysis tasks.
Undefined entries usually happen at poles. A denominator becomes zero there, so the transform is singular and the graph intentionally breaks at that point.
The region of convergence is the set of s values where the Laplace integral converges. For causal exponentials, it usually lies to the right of the rightmost pole.
Time shifts model delays. In the transform domain, a delay multiplies the result by an exponential factor. That rule is very common in systems and signal analysis.
Yes. You can download the current result data as CSV and save a PDF summary that includes the main transform information and sample table.
Start with positive real values away from poles. Then widen the interval to study decay, growth, or singular behavior more clearly.
Yes. Laplace transforms turn many linear differential equations into algebraic forms. That makes initial value problems easier to analyze and solve.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.