# Laplace & Inverse Laplace Transform Solver

Compute Laplace transforms for time-domain functions and inverse Laplace transforms with clear step-by-step solutions shown.

---

- **Canonical URL:** https://dothecalculation.com/calculators/laplace-transform-calculator
- **Category:** Math calculators
- **Publisher:** Do The Calculation (https://dothecalculation.com)
- **Cost:** Free, no account or sign-up required
- **Privacy:** Runs entirely in the browser; inputs are never sent to a server
- **Methodology:** https://dothecalculation.com/methodology

---

## Laplace Transform Solver

Calculate Laplace transforms for time-domain functions and inverse Laplace transforms with step-by-step expansions.

- Common t-domain functions
- s-domain rational fraction solver
- Step-by-step partial fraction inversion

## Introduction to Laplace Transforms and s-Domain Analysis

The Laplace transform is an integral transform that converts a function of a real variable \(t\) (often representing time) into a function of a complex variable \(s\) (representing complex frequency). Named after the mathematician Pierre-Simon Laplace, this transform is a cornerstone of engineering, physics, and applied mathematics. Its primary power lies in its ability to simplify differential equations — which govern dynamic physical systems — into simple algebraic equations, which are much easier to solve.

In time-domain analysis, solving differential equations involving derivatives, initial values, and driving forces can be mathematically tedious. By applying the Laplace transform, differentiation in the time domain is replaced by multiplication by \(s\) in the frequency domain. This allows engineers to solve for the system's transfer function, analyze stability, and then apply the inverse Laplace transform to return the solution to the time domain.

This calculator computes both forward Laplace transforms for standard functions and inverse Laplace transforms for proper rational s-domain functions, displaying the step-by-step partial fraction decompositions used during inversion.

## How to Use This Calculator

For a forward transform, pick a time-domain function (\(e^{at}\), \(\sin(at)\), \(\cos(at)\), or \(t\)) and its parameter \(a\). For an inverse transform, enter the numerator coefficients \(A\), \(B\) and denominator coefficients \(b\), \(c\) of \(F(s) = \frac{As+B}{s^2+bs+c}\). The calculator classifies the poles (distinct real, repeated real, or complex conjugate), applies the matching inversion method, and plots the resulting time-domain response.

## Mathematical Definition of Laplace and Inverse Transforms

The unilateral Laplace transform of a function \(f(t)\) defined for all real numbers \(t \geq 0\) is the integral \(F(s)\) defined as:

$$F(s) = \mathcal{L}\{f(t)\} = \int_{0}^{\infty} f(t) e^{-st}\, dt$$

Where \(s = \sigma + i\omega\) is a complex frequency parameter. The transform exists if the integral converges, which is true for functions of exponential order.

The inverse Laplace transform converts an s-domain function \(F(s)\) back into its time-domain counterpart \(f(t)\):

$$f(t) = \mathcal{L}^{-1}\{F(s)\} = \frac{1}{2\pi i} \lim_{T \to \infty} \int_{\gamma - iT}^{\gamma + iT} F(s) e^{st}\, ds$$

In practice, rather than evaluating this complex contour integral directly, we decompose \(F(s)\) using partial fractions and match the terms to a table of known transform pairs — this calculator's inverse mode does exactly that for a quadratic denominator.

## Worked Case Study: Inverting a Rational s-Domain Function

Let us find the inverse Laplace transform for the s-domain transfer function: $$F(s) = \frac{3s + 5}{s^2 - s - 2}$$

First, we factor the denominator: \(s^2 - s - 2 = (s - 2)(s + 1)\).

Next, we decompose \(F(s)\) into partial fractions:

$$\frac{3s + 5}{(s - 2)(s + 1)} = \frac{A}{s - 2} + \frac{B}{s + 1}$$

Solving for \(A\) and \(B\), we clear the denominators: \(3s + 5 = A(s + 1) + B(s - 2)\).

1. Set \(s = 2 \implies 11 = 3A \implies A = \frac{11}{3}\)

2. Set \(s = -1 \implies 2 = -3B \implies B = -\frac{2}{3}\)

This gives the decomposed form: $$F(s) = \frac{11}{3(s - 2)} - \frac{2}{3(s + 1)}$$

Now we apply the inverse transform to each term using the standard pair \(\mathcal{L}^{-1}\{\frac{1}{s-a}\} = e^{at}\):

$$f(t) = \frac{11}{3} e^{2t} - \frac{2}{3} e^{-t}$$

## System Stability and Pole-Zero Configurations

In control theory, the roots of the denominator of a transfer function \(F(s)\) are called the poles of the system, while the roots of the numerator are called the zeros. The positions of the poles in the complex s-plane determine the stability of the system. If any pole has a positive real part (lying in the right half of the s-plane), the system's time response will grow exponentially over time, indicating instability.

Conversely, if all poles lie in the left half of the s-plane, the system is stable, and its transient response will decay to zero. This calculator's inverse mode classifies the pole type (distinct real, repeated real, or complex conjugate) for whatever quadratic denominator you enter, and plots the resulting time response so you can see decay, growth, or oscillation directly.

## Related Calculators

The partial fraction decomposition used here is the same technique behind the [partial fractions calculator](/calculators/partial-fractions-calculator), useful on its own for integration problems. For the frequency-domain (rather than complex-frequency s-domain) view of a periodic signal, see the [Fourier series calculator](/calculators/fourier-series-calculator).

## Frequently asked questions

### What is a Laplace transform?

A Laplace transform is an integral transform that converts a time-domain function f(t) into a complex frequency-domain function F(s), simplifying differential equations into algebraic ones.

### What is the formula for a Laplace transform?

The Laplace transform of f(t) is L{f(t)} = ∫₀^∞ f(t) e^(−st) dt.

### What is the Laplace transform of a constant C?

The Laplace transform of a constant C is C/s, valid for s > 0. This calculator's forward mode doesn't include a constant-function preset, but it's a standard transform pair worth knowing.

### What is the Laplace transform of t^n?

The Laplace transform of tⁿ (for positive integers n) is n!/s^(n+1). This calculator's forward mode supports the linear case f(t) = t directly; higher powers aren't a preset option.

### What is the Laplace transform of e^(at)?

The Laplace transform of e^(at) is 1/(s − a), valid for s > a. This is one of the four preset forward-transform functions in this calculator.

### What is the Laplace transform of sin(at)?

The Laplace transform of sin(at) is a/(s² + a²).

### What is the Laplace transform of cos(at)?

The Laplace transform of cos(at) is s/(s² + a²).

### What is an inverse Laplace transform?

The inverse Laplace transform converts an s-domain function F(s) back into a time-domain function f(t), typically using partial fraction expansion and look-up tables.

### What form of F(s) does the inverse mode accept?

A proper rational function with a linear numerator and quadratic denominator, F(s) = (As + B)/(s² + bs + c). Higher-order denominators or numerators aren't currently supported.

### Why are poles important in Laplace transforms?

Poles are the roots of the denominator of F(s). Their values determine the exponential behavior of the system's response (e.g., decay, oscillation, or growth).

### What does system stability mean in the s-plane?

A system is stable if all its poles lie in the left half of the s-plane (negative real parts), which ensures the time-domain response decays to zero.

### How does differentiation in time affect the Laplace transform?

Differentiation in time corresponds to multiplication by s in the s-domain, minus the initial conditions: L{f'(t)} = sF(s) − f(0).

## Related concepts

- **Transfer Function** — A mathematical model representing the relationship between the input and output of a linear time-invariant system.
- **Partial Fractions** — An algebraic method used to break down complex rational expressions into simpler terms for easy integration or inversion.
- **Fourier Transform** — A related transform used to analyze periodic and non-periodic signals in the frequency domain, assuming no transient decay.

## Related guides

- [Scientific Notation Guide: Powers of Ten Made Practical](https://dothecalculation.com/blog/math/scientific-notation-basics) — Learn how to convert, compare, and calculate with powers of ten using worked examples and the live DTC scientific calculator.

## Related calculators

- [Derivative & Limit Solver](https://dothecalculation.com/calculators/derivative-solver) — Compute first and second derivatives of algebraic functions, find limits, and solve tangent line equations with clear steps.
- [Taylor & Maclaurin Series Calculator](https://dothecalculation.com/calculators/taylor-series-calculator) — Compute Taylor and Maclaurin polynomial series approximations with clear step-by-step error analysis and instant results.
- [Bisection & Newton-Raphson Root Finder](https://dothecalculation.com/calculators/bisection-method-calculator) — Find roots of nonlinear equations using interval bisection and Newton-Raphson iterative methods with this free numerical solver.
- [Boolean Algebra Simplifier & K-Map Solver](https://dothecalculation.com/calculators/boolean-algebra-calculator) — Simplify Boolean logic expressions, generate truth tables, and visualize Karnaugh Maps instantly with this free algebra solver.
- [Exponent Calculator](https://dothecalculation.com/calculators/exponent-calculator) — Solve exponent equations, simplify fractional powers into radicals, and compute large scientific notations with this free calculator.
- [Definite & Indefinite Integral Solver](https://dothecalculation.com/calculators/integral-solver) — Evaluate indefinite antiderivatives, definite integrals, and Riemann sum approximations instantly with this free calculus solver.

---

_This mathematical solver is designed for academic, engineering, and educational analysis. Rounding errors, numerical tolerances, or algorithm constraints might apply near poles, boundary conditions, or complex coordinate spaces. Always verify critical computations independently._

---

_Source: [Do The Calculation](https://dothecalculation.com/calculators/laplace-transform-calculator). Quote freely with attribution and a link to this page._