# Table of Laplace Transforms

**Laplace**

Laplace Transforms

# Rules

- linearity
- {$$af(t) + bg(t) \rightsquigarrow aF(s) + bG(s)$$}
*s*derivative- {$$ tf(t)\rightsquigarrow -F'(s)$$}

{$$ t^ nf(t) \rightsquigarrow (-1)^ nF^{(n)}(s)$$} *t*defivative- {$$f'(t) \rightsquigarrow sF(s)-f(0)$$}

{$$f^{(n)}(t)\rightsquigarrow s^ nF(s)-\left(f(0)s^{n-1}+f'(0)s^{n-2}+\cdots +f^{(n-1)}(0)\right)$$}

# Table

{$ f(t) $} | {$ F(s) $} | Conditions | * | |
---|---|---|---|---|

{$ 1 $} | {$\displaystyle \frac{1}{s} $} | {$ s>0 $} | Find the Laplace of 1 | |

{$ \displaystyle t $} | {$\displaystyle \frac{1}{s^2} $} | {$ \displaystyle \mathrm{Re}\, s > 0 $} | ||

{$\displaystyle t^2 $} | {$ \displaystyle \frac{2}{{{s}^{3}}} $} | {$\displaystyle \mathrm{Re}\, s > 0 $} | ||

{$\displaystyle t^n $} | {$\displaystyle \frac{n!}{s^{n+1}} $} | {$ \displaystyle \mathrm{Re}\, s> 0$} | {$n=0,1,2,\ldots$} | |

{$\displaystyle e^{rt} $} | {$\displaystyle \frac{1}{s-r} $} | {$ \displaystyle \mathrm{Re}\, s>\mathrm{Re}\, r $} | Find the Laplace of {$e^{rt}$} | exponential shift |

{$ \displaystyle \cos \omega t $} | {$ \displaystyle \frac{s}{s^2+\omega ^2} $} | {$\displaystyle \mathrm{Re}\, s>0 $} | ||

{$ \displaystyle \sin \omega t $} | {$ \displaystyle \frac{\omega }{s^2+\omega ^2} $} | {$ \displaystyle \mathrm{Re}\, s>0 $} | ||

{$ \displaystyle t\sin (\omega t) $} | {$\displaystyle \frac{2\omega s}{(s^2+\omega ^2)^2} $} | {$ $} | ||

{$ \displaystyle t\cos (\omega t) $} | {$\displaystyle \frac{s^2-\omega ^2}{(s^2+\omega ^2)^2} $} | {$ $} | ||

{$ \displaystyle e^{at}\cos (\omega t) $} | {$ \displaystyle \frac{s-a}{(s-a)^2+\omega ^2} $} | {$ \displaystyle \mathrm{Re}\, s>0 $} | ||

{$ \displaystyle e^{at}\sin (\omega t) $} | {$\displaystyle \frac{\omega }{(s-a)^2+\omega ^2} $} | {$\displaystyle \mathrm{Re}\, s>0 $} | ||

{$\displaystyle \frac{1}{t+1} $} | {$\displaystyle $} | {$ $} | Find the Laplace of {$ \frac{1}{t+1}$} | |

{$ $} | {$ $} | {$ $} |

<< | Working Laplace Trail | >>

*Sources:*

- Wiki Commons Believed to be in the public domain.

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** Categories:** Math Tables

** Tags:** Xxx Laplace Transforms

This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.

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