Inverse Laplace Transform Calculator

The inverse Laplace transform converts a function F(s) in the complex frequency domain back into a time-domain function f(t). Formally, if ℒ{f(t)} = F(s), then ℒ⁻¹{F(s)} = f(t). It is widely used in engineering and physics to solve differential equations by working in the s-domain and then converting the solution back to the time domain.

The Laplace transform takes a time-domain function f(t) and converts it to a frequency-domain function F(s), making differential equations easier to solve algebraically. The inverse Laplace transform does the reverse — it takes F(s) and recovers the original time-domain function f(t). Together they form a powerful pair for analyzing linear systems.

The standard approach is: (1) decompose F(s) into simpler fractions using partial fraction decomposition, (2) match each fraction to a known entry in the Laplace transform table, and (3) write down the corresponding time-domain terms. For example, 1/(s+a) corresponds to e^(−at), and ω/(s²+ω²) corresponds to sin(ωt).

The formal definition is the Bromwich integral: f(t) = (1/2πj) ∫ F(s)e^(st) ds, integrated along a vertical line in the complex plane. In practice, this contour integral is rarely computed directly — instead, tables of known transform pairs combined with partial fractions are used for virtually all engineering problems.

Type your expression using standard algebraic notation. Use * for multiplication (e.g. 2*s), ^ for exponents (e.g. s^2), and parentheses to group terms (e.g. (s+1)*(s+3)). For example, enter 5/(s^2+2*s+10) for the function 5/(s²+2s+10). You can also select a preset example from the dropdown to see common transform pairs.

The region of convergence is the set of values of s for which the Laplace transform integral converges. For a right-sided causal signal like e^(−at)u(t), the ROC is Re(s) > −a. The ROC is important for uniqueness — different time-domain functions can share the same F(s) expression but have different ROCs, corresponding to different signals.

Yes. Complex conjugate pole pairs produce damped sinusoidal terms in the time domain. For example, F(s) = 5/(s²+2s+10) has poles at s = −1 ± 3j, and its inverse Laplace transform is (5/3)e^(−t)sin(3t). The calculator handles these by completing the square and matching to standard sinusoidal transform pairs.

Key pairs include: 1/s → u(t) (unit step), 1/s² → t, n!/s^(n+1) → tⁿ, 1/(s+a) → e^(−at), ω/(s²+ω²) → sin(ωt), s/(s²+ω²) → cos(ωt), ω/((s+a)²+ω²) → e^(−at)sin(ωt), and (s+a)/((s+a)²+ω²) → e^(−at)cos(ωt). These cover the vast majority of functions encountered in engineering practice.