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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.