Laplace Transform Calculator. Enter a function f(t) and the Laplace Transform Calculator returns its transform F(s) using the standard definition ∫₀^∞ e^(−st)f(t)dt. Choose from common function types — exponential, trigonometric, polynomial, hyperbolic, and more — then select any shift or modifying parameters to get the corresponding transform formula and key properties displayed instantly. Also try the Summation Calculator (Sigma).
The Laplace transform converts a time-domain function f(t) into a complex frequency-domain function F(s) using the integral F(s) = ∫₀^∞ e^(−st) f(t) dt. It is widely used in engineering and physics to solve differential equations, analyze control systems, and study circuit behavior. See also our calculate Maclaurin Series Series Approximation at x.
The Laplace transform of e^(at) is F(s) = 1/(s − a), valid for s > a. This is one of the most fundamental transform pairs and forms the basis for many other results via the first shifting theorem.
The ROC is the set of values of s for which the Laplace integral converges. For example, for e^(at) the ROC is s > a, meaning the transform is only valid when the real part of s exceeds a. Identifying the ROC is essential for correctly interpreting F(s).
The Fourier transform evaluates F(jω) along the imaginary axis, while the Laplace transform uses a complex variable s = σ + jω and covers a broader region of convergence. This makes the Laplace transform more suitable for analyzing systems with initial conditions and transient behavior.
The Laplace transform of sin(ωt) is F(s) = ω / (s² + ω²), valid for s > 0. Similarly, the transform of cos(ωt) is s / (s² + ω²). These are standard results used in solving oscillatory differential equations.
The first shifting theorem states that if L{f(t)} = F(s), then L{e^(at) f(t)} = F(s − a). This allows you to compute transforms of exponentially modulated functions like e^(at)sin(ωt) by simply shifting the s-variable in the known result.
The Laplace transform of δ(t − a) is e^(−as), valid for all s. When a = 0, L{δ(t)} = 1, representing a flat spectrum. The Dirac delta is used to model instantaneous impulses in engineering systems.
Apply the Laplace transform to both sides of the equation, converting derivatives into algebraic terms (L{f'(t)} = sF(s) − f(0)). Solve the resulting algebraic equation for F(s), then apply the inverse Laplace transform to recover the time-domain solution f(t). You might also find our calculate Directional Derivative D_u f, Gradient ∂f/∂x at Point & Gradient ∂f/∂y at Point — Directional Derivative useful.