# What is the inverse Laplace of F(s)?

Finds f(t) from its Laplace transform F(s), by partial fractions and term by term, checked by numeric integration.

- Page: https://www.acalculator.org/math/inverse-laplace-transform-calculator
- JSON spec: https://www.acalculator.org/math/inverse-laplace-transform-calculator.json
- Version: 7848b966f35c

## Default answer

Example with the default inputs (Transform F(s) (3s + 2)/(s^2 - 3s + 2)): The inverse Laplace transform of (3s + 2)/(s^2 - 3s + 2) is f(t) = 8 e^(2t) - 5 e^t.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| F | Transform F(s) | A function of s, typed like (3s + 8)/(s^2 + 2s + 5) or 1/(s^2 - 1). |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| f | f(t) = | The function whose Laplace transform is F(s). |

## Method

F(s) is split into partial fractions; a computer algebra system inverts each one, and each result is checked: its Laplace transform, by numeric integration, must equal the fraction at three values of s.

## Assumptions

- f(t) is found for t ≥ 0.
- An answer that fails its check is not shown.

## Worked examples

1. F = (3s + 2)/(s^2 - 3s + 2) gives f = 8 e^(2t) - 5 e^t. Source: Trench, Elementary Differential Equations (2013), section 8.2 The Inverse Laplace Transform, Example 8.2.4. https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.02%3A_The_Inverse_Laplace_Transform.
2. F = 1/(s^2 - 1) gives f = e^t/2 - 1/(2 e^t). Source: Trench, Elementary Differential Equations (2013), section 8.2 The Inverse Laplace Transform, Example 8.2.1(a) (sinh t).
3. F = (s^2 - 5s + 7)/(s + 2)^3 gives f = 1/e^(2t) - 9t/e^(2t) + 21t^2/(2 e^(2t)). Source: Trench, Elementary Differential Equations (2013), section 8.2 The Inverse Laplace Transform, Example 8.2.7 (e^(−2t)(1 − 9t + 21t²/2)).

## FAQ

### What is the inverse Laplace transform?

The function f(t), for t ≥ 0, whose Laplace transform is F(s). It undoes the transform: if L[f] = F then L⁻¹[F] = f. For F(s) = 1/(s − 3), f(t) = e^(3t).

### How do I find an inverse Laplace transform?

Split F(s) into partial fractions and invert each with a table: 1/(s − a) gives e^(at), 1/(s − a)ⁿ gives t^(n−1) e^(at)/(n − 1)!, b/((s − a)² + b²) gives e^(at) sin(bt), and (s − a)/((s − a)² + b²) gives e^(at) cos(bt).

### Why use partial fractions?

A table lists only simple fractions. (3s + 2)/(s² − 3s + 2) is not in any table, but it equals −5/(s − 1) + 8/(s − 2), which inverts to −5eᵗ + 8e^(2t).

### What is the shift rule?

L[e^(at) f(t)] = F(s − a): multiplying f by e^(at) shifts its transform by a. Read backwards, the inverse of F(s − a) is e^(at) times the inverse of F(s). The page uses it for fractions such as 1/(s + 1)³, the inverse of 1/s³ = t²/2 shifted, giving t²e^(−t)/2.

### Can the page invert e^(−2s)/s?

No. That is the transform of a unit step that switches on at t = 2, and the page has no step functions. It gives no answer for such a transform.

### How is the answer checked?

Each part f(t) is transformed back by numeric integration at three values of s and compared with its fraction, and the whole f(t) is transformed back at two values of s past the rightmost pole and compared with F to 1 part in a million. The partial fractions are compared with F at 20 points. If a check fails, the page says "No verified answer".

## Sources

- William F. Trench, Elementary Differential Equations with Boundary Value Problems (2013), section 8.2 The Inverse Laplace Transform (LibreTexts): https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.02%3A_The_Inverse_Laplace_Transform