# Table of Laplace Transforms of Elementary Functions

Below are some functions $f(t)$ and their Laplace transforms $F(s)$.

$f(t)$ | $F(s) = \mathcal{L} \left\{f(t)\right\}$ |

$1$ | $\dfrac{1}{s}$ |

$t$ | $\dfrac{1}{s^2}$ |

$t^2$ | $\dfrac{2}{s^3}$ |

$t^n, \, (n = 1, \, 2, \, 3\, ...)$ | $\dfrac{n!}{s^{n + 1}}$ |

$t^a, \, (a \, \text{ is positive})$ | $\dfrac{\Gamma(a + 1)}{s^{a + 1}}$ |

$e^{at}$ | $\dfrac{1}{s - a}$ |

$\sin bt$ | $\dfrac{b}{s^2 + b^2}$ |

$\cos bt$ | $\dfrac{s}{s^2 + b^2}$ |

$\sinh at$ | $\dfrac{a}{s^2 - a^2}$ |

$\cosh at$ | $\dfrac{s}{s^2 - a^2}$ |

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