Common Laplace Transform Pairs

Core Idea

Standard Laplace transforms include: L[1] = 1/s, L[tⁿ] = n!/sⁿ⁺¹, L[e^at] = 1/(s-a), L[sin(bt)] = b/(s²+b²), L[cos(bt)] = s/(s²+b²). Tables of these pairs are essential references. Combinations via linearity and shifting theorems yield transforms of more complex functions, including step functions, impulses, and piecewise-defined inputs.

Explainer

From the Laplace transform definition, you know that L[f(t)] = ∫₀^∞ e^−st f(t) dt converts a function of t into a function of s. Each entry in the transform table comes from computing this integral once, carefully, and then recording the result so you never have to do it again. Understanding a handful of derivations — rather than memorizing all entries blindly — is enough to reconstruct the table and extend it when you encounter unfamiliar functions.

The simplest entry is L[1] = 1/s. Compute: ∫₀^∞ e^−st · 1 dt = [−(1/s)e^−st]₀^∞ = 0 − (−1/s) = 1/s, valid for s > 0. The next step up is L[tⁿ] = n!/sⁿ⁺¹, derived by repeated integration by parts (or by recognizing the gamma function integral). For n = 1: ∫₀^∞ t e^−st dt = 1/s². For general n, each integration by parts pulls down a factor of n and increases the power of s in the denominator, yielding n! in the numerator. The exponential L[e^at] = 1/(s−a) follows from the same first integral with s replaced by s−a, valid for s > a. Trig transforms follow from Euler's formula: e^ibt = cos(bt) + i sin(bt), so L[e^ibt] = 1/(s − ib); separating real and imaginary parts gives L[cos(bt)] = s/(s² + b²) and L[sin(bt)] = b/(s² + b²).

Once you have the basic pairs, two theorems extend them enormously. Linearity means L[af + bg] = aL[f] + bL[g]: you can transform linear combinations term by term, just like differentiation and integration. The first shifting theorem (s-shift) says L[e^atf(t)] = F(s−a) where F = L[f]: multiplying by an exponential in t shifts the transform variable s. So L[e^2tsin(3t)] = 3/((s−2)² + 9) simply by replacing s with s−2 in L[sin(3t)] = 3/(s² + 9). The second shifting theorem (t-shift) handles step functions: L[u_c(t) f(t−c)] = e^−cs F(s), where u_c is the Heaviside step function turning on at t = c. This lets you transform piecewise-defined functions that switch behavior at specific times — exactly what arises in engineering problems with switching inputs.

The real value of these tables emerges when solving ODEs. After transforming an initial-value problem, you obtain an algebraic expression in s for Y(s) = L[y(t)]. To recover y(t), you decompose Y(s) into a sum of recognizable table entries — often using partial fractions — and read off the inverse transform from the table. The transforms of derivatives (L[y'] = sY − y(0), L[y''] = s²Y − sy(0) − y'(0)) are how initial conditions enter. Knowing the common transform pairs fluently means you can match the pieces of a partial fraction decomposition to table entries immediately, turning the inverse transform step from a bottleneck into a mechanical lookup.