Elementary Laplace Transforms

$${\cal L}(1)(s) \frac{1}{s} \ \ (s>0)$$ = (1)

$${\cal L}(e^{at})(s) \frac{1}{s-a}\ \ (s>a)$$ = (2)

$${\cal L}(t^n)(s) \frac{n!}{s^{n+1}}\ \ (s>0,n \mbox{ a positive integer})$$ = (3)

$${\cal L}(t^p)(s) \frac{\Gamma(p+1)}{s^{p+1}}\ \ (s>0,p>-1)$$ = (4)

$${\cal L}(\sin(at))(s) \frac{a}{s^2+a^2}\ \ (s>0)$$ = (5)

$${\cal L}(\cos(at))(s) \frac{s}{s^2+a^2} \ \ (s>0)$$ = (6)

$${\cal L}(e^{at}\cdot \sin(bt))(s) \frac{b}{(s-a)^2+b^2}\ \ (s>a)$$ = (7)

$${\cal L}(e^{at}\cdot \cos(bt))(s) \frac{s-a}{(s-a)^2+b^2}\ \ (s>a)$$ = (8)

$${\cal L}(t^n\cdot e^{at})(s) \frac{n!}{(s-a)^{n+1}}\ \ (s>a)$$ = (9)

$${\cal L}(t^n \cdot f(t))(s) (-1)^n \frac{d^n}{ds^n}\left({\cal L}(f(t))\right)(s)$$ = (10)

$${\cal L}(f'(t))(s) s\cdot {\cal L}(f(t))(s) - f(0)$$ = (11)

$${\cal L}(H_c(t))(s) \frac{e^{-cs}}{s}\ \ (s>0)$$ = (12)

$${\cal L}(H_c(t)\cdot f(t-c))(s) e^{-cs}{\cal L}(f(t))(s)$$ = (13)

$${\cal L}(H_c(t)\cdot f(t))(s) {e^{-cs}{\cal L}(f(t+c))(s)}$$ = (14)

$${\cal L}(\delta_c(t))(s)$$ = e^-cs (15)

$${\cal L}(e^{ct}\cdot f(t))(s) {\cal L}(f(t))(s-c)$$ = (16)

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