Laplace Transform Calculator (Forward, Inverse, ODE)

Laplace transform calculator: standard pairs, inverse transform, linear ODE solving, initial/final value theorems and convolution f * g for LTI systems.

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Laplace transform

Laplace transform, introduced by Pierre-Simon Laplace in his Traite de Mecanique Celeste (1812), is a fundamental tool in mathematical analysis for solving linear differential equations, analyzing electrical circuits, control systems and vibration problems. Transforms f(t) of time variable t >= 0 into F(s) of complex variable s:

F(s) = L{f(t)} = integral_0^infinity e^(-st) * f(t) dt

Standard pairs

  • 1 -> 1/s (Re(s) > 0)
  • t^n -> n!/s^(n+1) (Re(s) > 0)
  • e^(at) -> 1/(s-a) (Re(s) > a)
  • sin(omega*t) -> omega/(s^2+omega^2)
  • cos(omega*t) -> s/(s^2+omega^2)
  • sinh(at), cosh(at) -> a/(s^2-a^2), s/(s^2-a^2)
  • delta(t) -> 1 (Dirac)
  • u(t-a) -> e^(-as)/s (Heaviside)
  • t*e^(at) -> 1/(s-a)^2
  • e^(at)*sin(omega*t) -> omega/((s-a)^2+omega^2)

Fundamental properties

  • Linearity: L{a*f + b*g} = a*F(s) + b*G(s).
  • Shift (Heaviside): L{u(t-a)*f(t-a)} = e^(-a*s)*F(s).
  • Damping: L{e^(at)*f(t)} = F(s-a).
  • Derivative: L{f'(t)} = s*F(s) - f(0).
  • Second derivative: L{f''(t)} = s^2*F(s) - s*f(0) - f'(0).
  • Integral: L{integral_0^t f(tau) dtau} = F(s)/s.
  • Convolution: L{(f*g)(t)} = F(s)*G(s).

Five modes

  1. Standard pair: selection of f(t) with its transform F(s) and ROC.
  2. Inverse: F(s) -> f(t) via table or partial fraction decomposition.
  3. ODE solution: application to linear differential equations.
  4. Initial/final value: limit theorems for stability analysis.
  5. Convolution: theorem L{f*g} = F(s)*G(s) for LTI systems.

Initial and final value theorems

Initial value theorem: f(0+) = lim_{s->infinity} s*F(s), if limit exists. Useful for initial system behavior without inverting.

Final value theorem: f(infinity) = lim_{s->0} s*F(s), valid only if all poles of s*F(s) have Re(s) < 0 (stable system). Gives steady-state response without solving.

Application to linear ODEs

Standard procedure to solve constant-coefficient linear ODEs:

  1. Apply L to both sides of equation.
  2. Use L{y'} = s Y(s) - y(0), L{y''} = s^2 Y(s) - s y(0) - y'(0), etc.
  3. Solve algebraically for Y(s).
  4. Decompose Y(s) into partial fractions.
  5. Apply L^-1 to elementary components via table.

Example: y'' + 2y' + 5y = 0, y(0)=1, y'(0)=0. Transform: (s^2 + 2s + 5) Y(s) - s - 2 = 0. Solving: Y(s) = (s+2)/(s^2 + 2s + 5). Inverse: y(t) = e^(-t) cos(2t) + (1/2) e^(-t) sin(2t).

Convolution theorem

One of the most useful properties: L{(f*g)(t)} = F(s)*G(s). Convolution (f*g)(t) = integral_0^t f(tau) g(t-tau) dtau. Transforms temporal convolution (complex operation) into product in Laplace domain. Applications:

  • LTI system response: y(t) = (h*x)(t) -> Y(s) = H(s) X(s).
  • Transfer function H(s): dynamical system characterization.
  • Volterra integral equations first and second kind.
  • Complex convolution via product of tabulated transforms.

Region of convergence (ROC)

Defining F(s) integral converges only for certain s values. ROC is set of complex s where integral converges absolutely. Typically right half-plane Re(s) > sigma_c, with sigma_c = 'abscissa of convergence'. ROC determines inverse transform uniqueness.

Applications

  • Electrical circuits: RLC transient analysis, active filters, transfer functions.
  • Control systems: stability analysis (Nyquist, Routh-Hurwitz).
  • Mechanics: mass-spring-damper vibrations.
  • Thermodynamics: heat diffusion, transient conduction.
  • Probability: characteristic functions, Laplace transforms of random variables.
  • Signal engineering: continuous signal processing, linear filters.
  • Mathematical physics: Volterra and Fredholm integral equations.

History of the Laplace transform

The transform is named after Pierre-Simon Laplace, the great French mathematician and astronomer who used its ideas in the 18th century in his work on probability theory. However, its modern use as a tool for solving differential equations is due above all to the English engineer Oliver Heaviside, who in the 19th century developed the so-called operational calculus to analyse electrical circuits. Heaviside's methods, though extraordinarily effective, were not mathematically rigorous; it was Thomas Bromwich, in the early 20th century, who provided a solid basis by connecting them to complex analysis and the contour inversion formula. Since then the Laplace transform has become one of the fundamental tools of applied mathematics, engineering and physics.

The Laplace transform in engineering

The Laplace transform is the natural language of systems and control engineering. Its power lies in transforming the differential equations of the time domain into simple algebraic equations in the domain of the complex variable s. In this domain, a linear system is described by its transfer function H(s), the ratio of the transform of the output to that of the input. The positions of the poles and zeros of H(s) in the complex plane determine the system's behaviour: stability is guaranteed if all the poles have negative real part. This approach underlies the design of regulators, filters, amplifiers and automatic control systems, from thermostats to aircraft autopilots. The transform converts complex dynamic problems into manageable algebraic manipulations.

Updated July 2026

Frequently Asked Questions

What is the Laplace transform?

It is a mathematical operation that converts a function of time f(t) into a function F(s) of a complex variable s, through the integral F(s) = ∫₀^∞ f(t)e^(−st)dt. Its main purpose is to transform differential equations, hard to solve in the time domain, into simple algebraic equations in the s domain, much easier to manipulate. It is fundamental in engineering, physics and systems theory.

Why is the Laplace transform useful for differential equations?

Because it transforms the operations of differentiation and integration into simple algebraic operations of multiplication and division by s. A linear differential equation thus becomes an algebraic equation in F(s), which is easily solved. Then, applying the inverse transform, one returns to the solution in the time domain. This method also automatically handles the initial conditions, incorporating them into the transform.

What is the transfer function?

It is the ratio H(s) of the Laplace transform of the output to that of the input of a linear system, with zero initial conditions. It completely characterises the system's behaviour: for any input, the output is obtained by multiplying its transform by H(s). The poles and zeros of H(s) determine the response and stability of the system, and it is the central tool of control theory.

What are the initial and final value theorems?

They are two shortcuts that let the behaviour of a function be known without computing the full inverse transform. The initial value theorem gives f(0⁺) = lim_{s→∞} s·F(s), the value of the function at the initial instant. The final value theorem gives f(∞) = lim_{s→0} s·F(s), the steady-state value, valid only if all poles of sF(s) have negative real part. They are very useful in stability analysis.

What is the convolution theorem?

It states that the Laplace transform of the convolution of two functions is the product of their transforms: L{(f∗g)(t)} = F(s)·G(s). The convolution (f∗g)(t) = ∫₀ᵗ f(τ)g(t−τ)dτ, complicated in the time domain, becomes a simple multiplication in the s domain. It is fundamental for analysing the response of a linear system to any input and for solving integral equations.

What is the region of convergence (ROC)?

It is the set of values of the complex variable s for which the integral defining the Laplace transform converges. For example, the transform of e^(at) is 1/(s−a) with ROC Re(s) > a. The ROC is essential to uniquely define the inverse transform: two different functions can have the same expression F(s) but different ROCs. In systems analysis, the ROC is linked to causality and stability.

What is the difference from the Fourier transform?

The Fourier transform uses e^(−iωt) and analyses the frequency content of a signal, while the Laplace transform uses e^(−st) with complex s, also including an exponential decay or growth component. Laplace is more general: it converges for a wider range of functions, handles initial conditions and is ideal for transients and causal systems, while Fourier is preferred for the frequency analysis of stationary signals.

Who developed the Laplace transform?

The ideas go back to Pierre-Simon Laplace in the 18th century, in the context of probability. The modern use for solving differential equations, however, is due to Oliver Heaviside, who in the 19th century developed the operational calculus for electrical circuits. Thomas Bromwich gave it a rigorous mathematical basis in the early 20th century. Today it is a standard tool in engineering, control theory, signal processing and mathematical physics.

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