# Table of Laplace Transformations

The following Table of Laplace Transforms is very useful when solving problems in science and engineering that require Laplace transform.

Each expression in the right hand column (the Laplace Transforms) comes from finding the infinite integral that we saw in the Definition of a Laplace Transform section.

Time Function `f(t)` `f(t)=mathcal(L)^-1{F(s)}` |
Laplace Transform of `f(t)``F(s)=mathcal(L){f(t)}` |
---|---|

1 | `1/s``s > 0` |

t (unit-ramp function) |
`1/s^2``s > 0` |

t^{n} (n, a positive integer) |
`(n!)/s^(n+1)``s > 0` |

e^{at} |
`1/(s-a)``s > a` |

sin ωt |
`omega/(s^2+omega^2)``s > 0` |

cos ωt |
`s/(s^2+omega^2)``s > 0` |

t^{n}g(t), for n = 1, 2, ... |
`(-1)^n (d^nG(s))/(ds^n)` |

t sin ωt |
`(2omegas)/((s^2+omega^2)^2)``s > |ω|` |

t cos ωt |
`(s^2-omega^2)/((s^2+omega^2)^2)``s > |ω|` |

`g(at)` | `1/a G (s/a)` Scale property |

`e^(at)g(t)` | `G(s − a)` Shift property |

e^{at}t^{n}, for n = 1, 2, ... |
`(n!)/((s-a)^[n+1])``s > a` |

te^{-t} |
`(1)/((s+1)^2)``s > -1` |

`1 − e^(-t//T)` | `(1)/(s(1+Ts))``s > -1/T` |

esin ω^{at}t |
`(omega)/((s-a)^2+omega^2)``s > a` |

ecos ω^{at}t |
`(s-a)/((s-a)^2+omega^2)` `s > a` |

`u(t)` | `1/s``s > 0` |

`u(t − a)` | `(e^[-as])/(s)``s > 0` |

`u(t − a) · g(t − a)` | e^{-as}G(s)
Time-displacement theorem |

g'(t) |
`sG(s) − g(0)` |

g''(t) |
`s^2 · G(s) − s · g(0) − g^'(0)` |

g^{(n)}(t) |
`s^n · G(s) ` `− s^(n-1) · g(0) ` `− s^(n-2) · g^'(0) −` ` ... − g^(n-1)(0)` |

`int_0^t g(t) \ dt` | `(G(s))/(s)` |

`int g(t)\ dt` | `(G(s))/(s)+1/s{intg(t) \ dt}_[t=0]` |

In the following sections we see how to use the Table of Laplace Transformations to solve problems.

Didn't find what you are looking for on this page? Try **search**:

### Online Algebra Solver

This algebra solver can solve a wide range of math problems. (Please be patient while it loads.)

Go to: Online algebra solver

### Ready for a break?

Play a math game.

(Well, not really a math game, but each game was made using math...)

### The IntMath Newsletter

Sign up for the free **IntMath Newsletter**. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!

### Share IntMath!

### Calculus Lessons on DVD

Easy to understand calculus lessons on DVD. See samples before you commit.

More info: Calculus videos