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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}$} 
{$ $}{$ $}{$ $}  

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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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