Laplace Transform Calculator

Compute forward and inverse Laplace transforms for common canonical signal families.

Scratchpad (not saved)

Save to create a persistent, shareable workspace.

What This Calculator Does

Compute forward and inverse Laplace transforms for common canonical signal families.

It combines Mode, Function Family, Polynomial Order n, a (exponential rate) to estimate Forward Transform, Sample Evaluation, Step.

Formula & Method

Core equations:

\mathcal{L}\{t^n\}=\frac{n!}{s^{n+1}},\;\mathcal{L}\{e^{at}\}=\frac{1}{s-a},\;\mathcal{L}\{\sin(bt)\}=\frac{b}{s^2+b^2},\;\mathcal{L}\{\cos(bt)\}=\frac{s}{s^2+b^2}

Notation used in the formulas: $R$ = Forward Transform; $x_{1}$ = Mode; $x_{2}$ = Function Family; $x_{3}$ = Polynomial Order n; $x_{4}$ = a (exponential rate); $x_{5}$ = b (angular frequency); $x_{6}$ = c (shift).

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use this calculator for quick scenario analysis. Start with baseline values, change one driver at a time, and compare how sensitive the results are to each input shown above.

Reference Book

Signals and Systems

Alan V. Oppenheim, Alan S. Willsky, and S. Hamid Nawab · Pearson

A classic reference for convolution, Fourier analysis, Laplace transforms, z-transforms, sampling, and LTI systems.

View Book

Inputs Used

Mode: Used directly in the calculation.
Function Family: Used directly in the calculation.
Polynomial Order n: Used directly in the calculation.
a (exponential rate): Used directly in the calculation.
b (angular frequency): Used directly in the calculation.
c (shift): Used directly in the calculation.
Evaluate at real s: Used directly in the calculation.
Evaluate inverse at t: Used directly in the calculation.

Related Learning

Use these guided proofs, examples, and explanations to go deeper on the ideas behind this tool.

Test convergence of an improper integral
An improper integral is decided by a limit, not by the antiderivative alone. This example tests $\int_1^{\infty} \frac{1}{x^2}\,dx$ and compares it with the divergent harmonic-tail integral.
What the definite integral means geometrically
The definite integral $\int_a^b f(x)\,dx$ measures signed area between a curve and the $x$-axis. This page builds that idea from Riemann sums, connects it to antiderivatives, and shows how to read integral notation.

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