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Calculator Answers

Laplace Transform Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A Laplace transform calculator is a mathematical tool that converts functions of a real variable (typically time, t) into a complex frequency-domain representation (s). It evaluates functions using a standard lookup table of transform pairs combined with linearity and shift theorem properties. This integral transform simplifies system analysis by turning differential equations into linear algebraic equations. Engineers and physicists use it to analyze electrical circuits, control systems, and transient state responses in physical systems.

Compute the Laplace transform F(s) of common functions of time f(t). View step-by-step shifts and formulas using standard transform pairs.

See also: Derivative Calculator, Integral Calculator, and Taylor Series Calculator.

Quick Answer

Compute the Laplace transform of standard time-domain functions. Enter your function f(t) to find its s-domain transform F(s) using transform pairs.

Supports common functions in variable t.

Try Examples

Step-by-step Explanation

For f(t) = sin(2t), the coefficient a = 2, giving the transform 2 / (s^2 + 4).
Laplace Transform

F(s) = L{f(t)} =

2/(s² + 4)

Formula: L{sin(at)} = a / (s² + a²)

Function f(t)sin(2t)
Transform F(s)2/(s² + 4)
Standard PairL{sin(at)} = a / (s² + a²)

Notice: This calculator uses standard pairs and linearity for common functions. For general symbolic Laplace transforms, refer to full CAS software.

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Examples

sin(2t)

2 / (s² + 4)

e^(3t)

1 / (s - 3)

t²

2 / s³

cos(5t)

s / (s² + 25)

e^(-2t) * sin(3t)

3 / ((s + 2)² + 9)

How it works

The calculator normalizes the function f(t) and checks it against a lookup table of standard Laplace transform pairs. It also applies the linearity property to handle sums of terms.

Integral Definition

F(s) = L{f(t)} = ∫₀^∞ e⁻ˢᵗ f(t) dt

Common Pairs

  • L{1} = 1/s
  • L{t} = 1/s²
  • L{eat} = 1/(s − a)
  • L{sin(at)} = a/(s² + a²)
  • L{cos(at)} = s/(s² + a²)

Applications in Control Systems & Engineering

In electrical engineering and control systems, the Laplace transform is used to analyze transfer functions of circuits and mechanical models. It helps determine stability, transient response, and frequency response without solving complex differential equations in the time domain.

Common Limitations

Not all functions have a Laplace transform. The integral definition requires that f(t) does not grow faster than exponentials as t approaches infinity (it must be of exponential order). For functions that grow extremely fast, like et², the Laplace integral does not converge.

Related Mathematical Tools

If you are working on numerical double integration, visit our Double Integral Calculator. For derivatives or rate of change analyses, check out our Rate of Change Calculator.

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Derivative CalculatorDifferentiate polynomials term by term with the power rule — clear breakdown of each term's derivative.Integral CalculatorCompute definite integrals of polynomials with the power rule, or numerically with Simpson's rule for safe expressions.Taylor Series CalculatorApproximate any differentiable function f(x) with a Taylor polynomial centered at a specified point a with successive derivatives.Partial Derivative CalculatorCalculate the partial derivative of any multi-variable function with respect to a chosen variable with optional point evaluation.

Frequently asked questions

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).

The Laplace transform of f(t) is defined by the improper integral: F(s) = ∫₀^∞ e^(-st) f(t) dt, for s > 0.

It is widely used to solve linear differential equations. By converting differential operations in the time domain into simple algebraic operations in the s-domain, it simplifies system analysis in engineering and physics.

The Linearity Property states that the Laplace transform of a sum of functions scaled by constants is equal to the sum of their individual Laplace transforms: L{a*f(t) + b*g(t)} = a*L{f(t)} + b*L{g(t)}.

The First Shift Theorem states that multiplying a function by an exponential eat shifts its Laplace transform in the s-domain: L{eat f(t)} = F(s − a), where F(s) is the Laplace transform of f(t).