Mathematics · Calculus · Complex Analysis
Laplace Transform Calculator
Compute the Laplace transform of common functions using the integral definition and standard transform pairs.
Calculator
Formula
f(t) is the time-domain function defined for t ≥ 0. s = σ + jω is the complex frequency variable (σ, ω ∈ ℝ). F(s) is the resulting transform in the s-domain. The integral converges when σ = Re(s) is greater than the abscissa of absolute convergence for f(t).
Source: Kreyszig, E. — Advanced Engineering Mathematics, 10th Ed., Wiley (2011); Abramowitz & Stegun, Chapter 29.
How it works
The one-sided (unilateral) Laplace transform is defined by the integral F(s) = ∫₀^∞ e^(−st) f(t) dt, where s = σ + jω is the complex frequency variable. By multiplying f(t) by the decaying exponential e^(−st) before integrating, the transform maps information about how a signal grows or oscillates over time into algebraic expressions in s. This is analogous to how the Fourier transform captures frequency content, but the Laplace transform handles a broader class of signals including unstable or growing ones, provided Re(s) is large enough.
Each function has a closed-form transform pair. For example, a unit step u(t) = 1 transforms to F(s) = 1/s, an exponential e^(at) transforms to 1/(s−a) valid for Re(s) > a, and sin(ωt) transforms to ω/(s² + ω²). The region of convergence (ROC) specifies where in the complex s-plane the integral converges absolutely — a critical detail for stability analysis and inverse transforms. Linearity is a key property: the transform of a sum is the sum of transforms, and constants factor out freely.
In practice, engineers rarely evaluate the defining integral directly. Instead, they consult transform tables (like those in Kreyszig or Abramowitz & Stegun), apply linearity and shift theorems, and compose transforms from known pairs. The First Shifting Theorem states that if F(s) = ℒ{f(t)}, then ℒ{e^(at)f(t)} = F(s−a), which explains why damped sinusoids share the same form as pure sinusoids but shifted in s. Once a problem is solved algebraically in the s-domain, the inverse Laplace transform (using partial fractions and tables) recovers the time-domain solution.
Worked example
Problem: Find the Laplace transform of f(t) = e^(−2t)·sin(3t) and evaluate it at s = 5.
Step 1 — Identify the function type. This is a damped sine: f(t) = e^(at)·sin(ωt) with a = −2 and ω = 3.
Step 2 — Apply the transform pair. The standard result for the damped sine is: F(s) = ω / [(s−a)² + ω²]. Substituting: F(s) = 3 / [(s − (−2))² + 3²] = 3 / [(s + 2)² + 9].
Step 3 — State the region of convergence. The ROC is Re(s) > a = −2, meaning the transform converges for any s with real part greater than −2.
Step 4 — Evaluate at s = 5. F(5) = 3 / [(5 + 2)² + 9] = 3 / [49 + 9] = 3 / 58 ≈ 0.051724.
Interpretation: The value F(5) ≈ 0.0517 represents the weight of the e^(−5t) component when decomposing f(t) into complex exponentials. This kind of evaluation is essential when checking residues in partial fraction expansions or verifying Laplace-domain filter responses.
Limitations & notes
This calculator covers the ten most common unilateral Laplace transform pairs and does not support arbitrary symbolic input or piecewise functions. The numerical evaluation of F(s) assumes s is real; for complex s, manual computation of both real and imaginary parts is required. The power function t^n is restricted to non-negative integer n, since the factorial formula n!/s^(n+1) applies only in that case (fractional powers require the Gamma function: Γ(n+1)/s^(n+1)). For hyperbolic functions sinh(ωt) and cosh(ωt), the ROC requires Re(s) > |ω|; evaluating at |s| ≤ |ω| causes a pole and the calculator returns NaN to signal divergence. The Laplace transform assumes f(t) = 0 for t < 0 (causal signals); non-causal signals require the bilateral Laplace transform, which is not covered here. Always verify that your chosen s lies within the ROC before trusting a numerical result.
Frequently asked questions
What is the difference between the Laplace transform and the Fourier transform?
The Fourier transform evaluates the integral along the imaginary axis (s = jω), while the Laplace transform uses the full complex plane s = σ + jω. This makes the Laplace transform applicable to a wider class of functions, including exponentially growing signals, and introduces the concept of the region of convergence. When σ = 0 and the ROC includes the imaginary axis, the Laplace and Fourier transforms coincide.
What does the region of convergence (ROC) mean physically?
The ROC is the set of complex values s for which the Laplace integral converges absolutely. For causal, stable systems, the ROC is a right-half plane Re(s) > σ₀. The ROC determines uniqueness of the inverse transform and encodes stability information: if the ROC includes the imaginary axis, the system is bounded-input bounded-output (BIBO) stable.
How do I use the First Shifting Theorem in practice?
The First Shifting Theorem states: ℒ{e^(at)·f(t)} = F(s−a), where F(s) = ℒ{f(t)}. To use it, first find the transform of f(t) alone, then replace every s with (s−a). For example, ℒ{sin(ωt)} = ω/(s²+ω²), so ℒ{e^(at)sin(ωt)} = ω/((s−a)²+ω²). This is exactly the damped sine pair used in this calculator.
Can the Laplace transform handle initial conditions for ODEs?
Yes — this is one of its greatest strengths. When transforming a derivative, ℒ{f′(t)} = sF(s) − f(0⁻), automatically incorporating the initial condition f(0⁻). For a second derivative, ℒ{f″(t)} = s²F(s) − sf(0⁻) − f′(0⁻). This converts an ODE with initial conditions into an algebraic equation in F(s), which can be solved and then inverted.
What is the inverse Laplace transform and how is it computed?
The inverse Laplace transform recovers f(t) from F(s) via the Bromwich integral: f(t) = (1/2πj)∫_{σ−j∞}^{σ+j∞} e^(st) F(s) ds. In practice, engineers use partial fraction decomposition to express F(s) as a sum of recognizable transform pairs, then read f(t) directly from a table. For example, 3/[(s+2)²+9] immediately reads as e^(−2t)sin(3t) from the damped-sine pair.
Last updated: 2025-01-15 · Formula verified against primary sources.