Inverse Laplace Transform Calculator

Inverse Laplace Transform Calculator. Enter a function F(s) in the frequency domain and get back its inverse Laplace transform f(t). Type your expression using standard notation — for example 5/(s^2+2s+10) or 1/(s*(s+1)) — and the calculator applies partial fraction decomposition and known transform pairs to return the time-domain result. Supports common forms including polynomials, exponentials, sinusoids, and step functions. Also try the Double Integral Calculator.

Function F(s) *

Enter your frequency-domain function. Use * for multiplication, ^ for powers. Examples: 1/s, 3/(s+2), s/(s^2+4)

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Results

Inverse Laplace Transform f(t)

Transform Type

Method Applied

Region of Convergence

Results Table

What is the inverse Laplace transform?

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. See also our calculate Average Rate of Change (Δy/Δx), Change in y (Δy = y₂ − y₁) & Change in x (Δx = x₂ − x₁) — Average Rate of Change.

What is the difference between the Laplace transform and the inverse Laplace transform?

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.

How do I find the inverse Laplace transform of a function?

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).

What is the inverse Laplace transform formula?

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.

How do I enter functions into this calculator?

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.

What is the region of convergence (ROC)?

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.

Can this calculator handle functions with complex poles?

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.

What are the most common inverse Laplace 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. You might also find our calculate Derivative f′(x), Outer Function g(u) & Inner Function u = h(x) — Chain Rule Derivative useful.