# What is the Laplace transform of f(t)?

Finds the Laplace transform F(s) of a function f(t), term by term, as partial fractions, checked by numeric integration.

- Page: https://www.acalculator.org/math/laplace-transform-calculator
- JSON spec: https://www.acalculator.org/math/laplace-transform-calculator.json
- Version: 94b386dd9de4

## Default answer

Example with the default inputs (Function f(t) t^3 e^(4t)): The Laplace transform of t^3 e^(4t) is F(s) = 6/(s - 4)^4.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Function f(t) | A function of t, typed like t^2 e^(3t), sin(2t) or 3t^4 - 2t + 5. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| F | F(s) = | The Laplace transform: the integral of e^(−st) f(t) from 0 to ∞, for s larger than the growth rate of f. |

## Method

F(s) = ∫ from 0 to ∞ of e^(−st) f(t) dt. A computer algebra system transforms each term; each transform is checked against numeric integration at three values of s.

## Assumptions

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

## Worked examples

1. f = t^3 e^(4t) gives F = 6/(s - 4)^4. Source: Trench, Elementary Differential Equations (2013), section 8.1 Introduction to the Laplace Transform, Example 8.1.5. https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.01%3A_Introduction_to_the_Laplace_Transform.
2. f = sin(2t) gives F = 2/(4 + s^2). Source: Trench, Elementary Differential Equations (2013), section 8.1 Introduction to the Laplace Transform, Example 8.1.4 (ω = 2: ω/(s² + ω²)).
3. f = cosh(3t) gives F = 1/(2 (s - 3)) + 1/(2 (3 + s)). Source: Trench, Elementary Differential Equations (2013), section 8.1 Introduction to the Laplace Transform, Example 8.1.6 (b = 3: s/(s² − 9), in partial fractions).

## FAQ

### What is the Laplace transform?

It turns a function of time f(t), for t ≥ 0, into a function of s: F(s) = ∫ from 0 to ∞ of e^(−st) f(t) dt. Derivatives in t become multiplication by s, which turns linear differential equations into algebra.

### What are the most common transforms?

L[1] = 1/s, L[tⁿ] = n!/s^(n+1), L[e^(at)] = 1/(s − a), L[sin(bt)] = b/(s² + b²), L[cos(bt)] = s/(s² + b²), L[tⁿ e^(at)] = n!/(s − a)^(n+1), and L[e^(at) sin(bt)] = b/((s − a)² + b²). Each holds for s larger than the growth rate of f.

### How does the page handle a sum like 3t⁴ − 2t + 5?

The transform is linear, so it transforms each term and adds: 3 · 24/s⁵ − 2/s² + 5/s = 72/s⁵ − 2/s² + 5/s.

### For which s does F(s) hold?

For s larger than the exponential growth rate of f: s > 0 for 1, t and sin(2t); s > 4 for t³e^(4t). The page gives the formula; the integral diverges for smaller s.

### Why is there no transform of 1/t?

The integral ∫ from 0 of e^(−st)/t dt is infinite near t = 0 for every s, so 1/t has no Laplace transform. Functions that grow faster than any exponential, such as e^(t²), have none either.

### How is the answer checked?

Each term's transform is compared with a numeric integral of e^(−st) f(t) at three values of s. The partial-fraction form is compared with the sum it rewrites 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 8.1 Introduction to the Laplace Transform (LibreTexts): https://math.libretexts.org/Bookshelves/Differential_Equations/Elementary_Differential_Equations_with_Boundary_Value_Problems_(Trench)/08%3A_Laplace_Transforms/8.01%3A_Introduction_to_the_Laplace_Transform