Laplace Transform Calculator

Step-by-Step Forward & Inverse Symbolic CAS Free · No Signup

Compute forward and inverse Laplace transforms with detailed step-by-step solutions. Features partial fraction decomposition, region of convergence, a complete Laplace pairs reference table, 2D graphs, and a built-in Python compiler. Essential for differential equations, control systems, and signal processing.

Laplace Transform
Enter a function of t
Type an expression above…
f(t)F(s)ROC
Re(s) > 0
Re(s) > 0
Re(s) > −a
Re(s) > 0
Re(s) > 0
Re(s) > −a
Re(s) > −a
all s
Re(s) > 0
t^2    s^3    (s+1)^2
sin(2*t)    cos(5*t)
e^(-3*t)    exp(-t)
sinh(t)    cosh(t)
Heaviside(t-2)    DiracDelta(t)
1/(s^2+1)    s/(s^2+9)
Multiplication: Use * explicitly: t*e^(-t) not te^(-t)
Powers: t^2 or (s+1)^2
Constants: pi, e

Result

ℒ

Enter an expression and click Compute

Compute forward or inverse Laplace transforms with step-by-step solutions.

f(t) vs t

Compute a transform to see the time-domain plot.

Python Compiler

What is the Laplace Transform?

The Laplace transform is an integral transform that converts a function of time f(t) into a function of complex frequency F(s). It is defined as L{f(t)} = ∫0∞ f(t) e−st dt, where s = σ + jω is a complex variable.

The transform is essential in engineering and physics for solving ordinary differential equations, analyzing linear time-invariant (LTI) systems, and designing control systems and electrical circuits.

Key Properties

Linearity

L{af(t) + bg(t)} = aF(s) + bG(s). The transform distributes over addition and scalar multiplication.

First Shifting Theorem

L{e−atf(t)} = F(s+a). Multiplication by an exponential shifts the s-domain by a.

Derivative Property

L{f'(t)} = sF(s) − f(0). Differentiation becomes multiplication by s, converting ODEs to algebra.

Convolution Theorem

L{f*g} = F(s)·G(s). Convolution in the time domain becomes multiplication in the s-domain.

Applications

⚡

Circuit Analysis

Analyze RLC circuits by transforming differential equations into algebraic equations in the s-domain.

⚙

Control Systems

Transfer functions H(s) describe system input-output relationships. Used for stability and feedback analysis.

📈

Signal Processing

Filter design and frequency response analysis of continuous-time linear systems.

📚

Differential Equations

Solve initial-value problems by converting ODEs to algebraic equations, solving for F(s), then inverting.

Frequently Asked Questions

The Laplace transform converts a time-domain function f(t) into a complex frequency-domain function F(s) using the integral L{f(t)} = integral from 0 to infinity of f(t)*e^(-st) dt. It is widely used to solve differential equations, analyze control systems, and study circuit behavior.
The inverse Laplace transform converts F(s) back to f(t). The most common method is partial fraction decomposition: break F(s) into simpler fractions, then look up each fraction in a table of known Laplace pairs. For example, 1/(s+a) transforms back to e^(-at).
The region of convergence is the set of complex values of s for which the Laplace transform integral converges. For causal signals (t >= 0), the ROC is a right half-plane Re(s) > sigma. For example, L{e^(-at)} = 1/(s+a) with ROC Re(s) > -a.
Use partial fractions when F(s) is a rational function (ratio of polynomials) that does not directly match a standard Laplace pair. Decomposing into simpler terms like A/(s+a) + B/(s+b) makes it easy to look up each term in the transform table.
The Heaviside step function theta(t) equals 0 for t < 0 and 1 for t >= 0. In Laplace transform results, it indicates that the solution is valid only for t >= 0. SymPy includes it automatically since the Laplace transform is defined for causal (one-sided) signals.
Yes, this calculator is completely free with no signup required. You get symbolic computation via SymPy, step-by-step solutions, 2D graphs, LaTeX export, and a built-in Python compiler.

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