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# Laplace Transforms

### Laplace Transformations.

Term | Definition |
---|---|

L [1] | 1/s |

L [ t^n ] | (n!) / [s^(n+1)] |

L [sin (at)] | a/ [ s^2+a^2 ] |

L [cos (at)] | (s-a)/ [ s^2+a^2 ] |

L [ sinh (at) ] | a/ [ s^2 - a^2 ] |

L [cosh (at)] | (s-a)/ [ s^2 - a^2 ] |

L [e^(at) sin (bt)] | a/ [ (s-a)^2+b^2 ] |

L [e^(at) cos (bt)] | (s-a)/ [ (s-a)^2+b^2 ] |

L [e^(at) sinh (bt)] | a/ [ (s-a)^2- b^2 ] |

L [e^(at) cosh (bt)] | (s-a)/ [ (s-a)^2- b^2 ] |

L [ t^(n ) * e^(at) ] | (n!) / [ (s-a)^(n+1)] |

L [U_c (t)] = L [U (t-c)] Heaviside Function | [e^(-cs) ]/s |

L[U_c (t) * f(t-c) ] | [e^(-cs) * F(s)] where F(s) is the integral of f(s) in terms of s |

L[e^(ct) * f(t) ] | F(s-c) where F(s) is the integral of f(s) in terms of s |

L[ (1/t)* f(t) ] | Integral of F(u) in terms of u with top bound infinity to bottom bound s. |

L [f(ct)] | (1/c)*F(s/c) |

L [( δ-c)] Dirac Delta Function | e^(-cs) |