# 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.

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