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Home › Math & Conversions › Core Math & Algebra › Laplace transform Laplace Transform Calculator Compute Laplace transforms for standard time-domain functions, see the formulas and region of convergence (ROC), and get numeric values at a chosen real s 0 . Core Math & Algebra Interactive Laplace transform tool Choose a standard time-domain signal \(f(t)\), set its parameters, and let the calculator produce the symbolic Laplace transform \(F(s)\), with a short explanation and region of convergence. Optionally, evaluate \(F(s)\) at a specific real value \(s_0\). Standard Laplace pairs Custom ODE solver (coming soon) Time-domain function \(f(t)\) Exponential: \(f(t) = K e^{a t} u(t)\) Power: \(f(t) = K t^n u(t)\) Sine: \(f(t) = K \sin(b t) u(t)\) Cosine: \(f(t) = K \cos(b t) u(t)\) Damped sine: \(f(t) = K e^{a t} \sin(b t) u(t)\) Damped cosine: \(f(t) = K e^{a t} \cos(b t) u(t)\) Shifted: \(f(t) = K u(t-a_0) g(t-a_0)\) Impulse: \(f(t) = K \delta(t-a_0)\) All signals are assumed causal (zero for \(t < 0\)) unless explicitly shifted by \(a_0\). Optional numeric evaluation \(s_0\) (real) If provided and within the ROC, the tool evaluates \(F(s_0)\) numerically. Amplitude \(K\) Real gain (can be negative). Exponential rate \(a\) Real \(a\). For a stable decaying exponential, use \(a < 0\). Notes \(f(t) = K e^{a t} u(t)\) → \(F(s) = \dfrac{K}{s - a}\), ROC: \(\mathrm{Re}(s) > \mathrm{Re}(a)\). Amplitude \(K\) Real gain. Power \(n\) (integer \(\ge 0\)) Common cases: \(n = 0,1,2,\dots\). Notes \(f(t) = K t^n u(t)\) → \(F(s) = K \dfrac{n!}{s^{n+1}}\), ROC: \(\mathrm{Re}(s) > 0\). Amplitude \(K\) Angular frequency \(b > 0\) \(b\) in rad/s (or the same unit as \(t^{-1}\)). \(f(t) = K \sin(bt) u(t)\) → \(F(s) = K \dfrac{b}{s^2 + b^2}\), ROC: \(\mathrm{Re}(s) > 0\). Amplitude \(K\) Angular frequency \(b > 0\) \(f(t) = K \cos(bt) u(t)\) → \(F(s) = K \dfrac{s}{s^2 + b^2}\), ROC: \(\mathrm{Re}(s) > 0\). Amplitude \(K\) Exponential rate \(a\) Angular frequency \(b > 0\) \(K e^{at} \sin(bt) u(t) \rightarrow F(s) = K \dfrac{b}{(s-a)^2 + b^2}\), ROC: \(\mathrm{Re}(s) > \mathrm{Re}(a)\). Amplitude \(K\) Exponential rate \(a\) Angular frequency \(b > 0\) \(K e^{at} \cos(bt) u(t) \rightarrow F(s) = K \dfrac{s-a}{(s-a)^2 + b^2}\), ROC: \(\mathrm{Re}(s) > \mathrm{Re}(a)\). Amplitude \(K\) Shift \(a_0 \ge 0\) Start time of the shifted signal. \(G(s)\) (Laplace of \(g(t)\)) Symbolic placeholder for the base transform \(G(s)\). If \(f(t) = K u(t-a_0) g(t-a_0)\) and \(\mathcal{L}\{g(t)\} = G(s)\), then \(\mathcal{L}\{f(t)\} = K e^{-a_0 s} G(s)\) with ROC equal to that of \(G(s)\). Amplitude \(K\) Shift \(a_0 \ge 0\) \(\mathcal{L}\{K \delta(t-a_0)\} = K e^{-a_0 s}\), ROC: all complex \(s\). Compute Laplace transform Clear Load example: damped sine Result summary Time-domain signal \(f(t)\) Laplace transform \(F(s)\) Region of convergence (ROC) Numeric evaluation at \(s_0\) Definition of the Laplace transform The unilateral Laplace transform of a time-domain function \(f(t)\), assumed zero for \(t < 0\), is defined as \(\displaystyle \mathcal{L}\{f(t)\}(s) = F(s) = \int_{0^-}^{\infty} f(t)\,e^{-st}\,\mathrm{d}t\) Here \(s\) is a complex variable, \(s = \sigma + j\omega\). For some functions the integral converges only for a subset of the complex plane, called the region of convergence (ROC). In control, circuits, and mechanical systems, the Laplace transform is a standard tool to turn differential equations into simple algebra in the \(s\)-domain, where transfer functions and poles/zeros are easier to interpret. Standard Laplace transform pairs The table below collects a few core Laplace transform pairs used most often in introductory courses, signal processing, and control engineering. \(f(t)\) \(F(s) = \mathcal{L}\{f(t)\}\) ROC \(u(t)\) \(\dfrac{1}{s}\) \(\mathrm{Re}(s) > 0\) \(e^{at} u(t)\) \(\dfrac{1}{s - a}\) \(\mathrm{Re}(s) > \mathrm{Re}(a)\) \(t^n u(t)\), \(n \in \mathbb{N}_0\) \(\dfrac{n!}{s^{n+1}}\) \(\mathrm{Re}(s) > 0\) \(\sin(bt) u(t)\) \(\dfrac{b}{s^2 + b^2}\) \(\mathrm{Re}(s) > 0\) \(\cos(bt) u(t)\) \(\dfrac{s}{s^2 + b^2}\) \(\mathrm{Re}(s) > 0\) \(e^{at}\sin(bt) u(t)\) \(\dfrac{b}{(s-a)^2 + b^2}\) \(\mathrm{Re}(s) > \mathrm{Re}(a)\) \(e^{at}\cos(bt) u(t)\) \(\dfrac{s-a}{(s-a)^2 + b^2}\) \(\mathrm{Re}(s) > \mathrm{Re}(a)\) \(u(t-a)f(t-a)\) \(e^{-as} F(s)\) Same ROC as \(F(s)\) \(\delta(t-a)\) \(e^{-as}\) All \(s\) ROC and stability intuition The region of convergence encodes how quickly the signal decays (or grows) and determines whether the Laplace transform exists. For example, a causal exponential \(e^{at}u(t)\) only has a finite transform if \(e^{(a-\sigma)t}\) decays as \(t \rightarrow \infty\), which requires \(\mathrm{Re}(s) = \sigma > \mathrm{Re}(a)\). In systems theory, the ROC relates directly to stability : if all poles of a causal LTI system lie in the left half-plane (\(\mathrm{Re}(s) < 0\)), then the ROC includes the imaginary axis and the system is BIBO stable. Laplace transform – FAQ Why is the Laplace transform so useful for differential equations? The Laplace transform converts differentiation into multiplication by \(s\): \(\mathcal{L}\{f'(t)\} = sF(s) - f(0^-)\). Higher-order derivatives lead to higher powers of \(s\). This turns linear constant-coefficient differential equations into algebraic equations in \(s\), which are much easier to solve; you then apply the inverse transform to return to \(f(t)\). What is the difference between unilateral and bilateral Laplace transforms? The unilateral (one-sided) Laplace transform integrates from \(0^-\) to \(\infty\) and is tailored to causal signals and initial-value problems. The bilateral (two-sided) transform integrates from \(-\infty\) to \(\infty\) and is useful when signals have non-zero values for negative time. This calculator follows the common unilateral convention. Can I combine results for sums of signals? Yes. The Laplace transform is linear: \(\mathcal{L}\{a f(t) + b g(t)\} = a F(s) + b G(s)\). You can decompose a complicated \(f(t)\) into a sum of standard building blocks, apply the table or the calculator to each term, then add the corresponding \(F(s)\) expressions. How accurate are the numeric evaluations? The numeric values use standard double-precision floating point, comparable to scientific calculators and engineering software. For extreme parameter values (very large or small magnitudes) rounding and overflow may occur; these limitations are inherent to finite-precision arithmetic rather than the transform formulas themselves. Frequently Asked Questions Do I always need the Laplace transform, or is the Fourier transform enough? The Fourier transform assumes signals are sufficiently well-behaved over all time and often focuses on steady-state sinusoidal behaviour. The Laplace transform generalizes Fourier by including exponential growth/decay and transient behaviour through the real part of s. In control and transient analysis, the Laplace transform is usually the more natural tool. Does this calculator compute inverse Laplace transforms? This version focuses on forward Laplace transforms (from \(f(t)\) to \(F(s)\)) with clear ROC descriptions. For many engineering problems, forward transforms combined with partial fraction decompositions and standard tables are sufficient. A future update may include a guided inverse-transform helper for simple rational \(F(s)\). How should I document Laplace transform steps in reports or lab work? When you use transform methods in formal work, always show the original differential equation, the initial conditions, the expression for \(F(s)\), and the corresponding inverse transform step. This calculator can help you verify each transform pair numerically, but your report should still include the full reasoning and references to standard tables. Related Core Math & Algebra tools Explore more tools that support calculus, algebra, and probability work alongside Laplace transforms. Convolution visualizer Pythagorean theorem SOHCAHTOA (trig ratios) Exponent rules Continued fraction Fibonacci number Dice roll probability Base converter Set theory basics Interval notation All Math & Conversions tools Checklist: using Laplace transforms safely Clarify whether you are using the unilateral or bilateral definition. Ensure \(f(t)\) is well-defined for \(t \ge 0\) and note any discontinuities or impulses. Write down the standard transform pair you are using and check its ROC. For sums of terms, apply linearity and transform each term separately. When solving ODEs, always apply the inverse transform and verify initial conditions are satisfied.
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