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 $$\mathcal{L} \{1\}(s)$$ = $$\dfrac{1}{s} \; (s>0)$$ (1) $$\mathcal{L} \{e^{at}\}(s)$$ = $$\dfrac{1}{s-a} \; (s>a)$$ (2) $$\mathcal{L} \{t^{n}\}(s)$$ = $$\dfrac{n!}{s^{n+1}} \;\; (s>0, n \text{ is a positive integer})$$ (3) $$\mathcal{L} \{t^{p}\}(s)$$ = 4) $$\mathcal{L} \{\sin (at)\}(s)$$ = $$\dfrac{a}{s^2+a^2} \; (s>0)$$ (5) $$\mathcal{L} \{\cos (at)\}(s)$$ = $$\dfrac{s}{s^2+a^2} \; (s>0)$$ (6) $$\mathcal{L} \{e^{at} \cdot \sin (bt)\}(s)$$ = $$\dfrac{b}{(s-a)^2+b^2} \; (s>a)$$ (7) $$\mathcal{L} \{e^{at} \cdot \cos (bt)\}(s)$$ = $$\dfrac{s-a}{(s-a)^2+b^2} \; (s>a)$$ (8) $$\mathcal{L} \{t^{n} \cdot e^{at}\}(s)$$ = $$\dfrac{n!}{(s-a)^{n+1}} \; (s>a)$$ (9) $$\mathcal{L} \{t^{n} \cdot f(t)\}(s)$$ = $$(-1)^n \dfrac{d^n}{ds^n} \mathcal{L} \{f(t)\}(s)$$ (10) $$\mathcal{L} \{f'(t)\}(s)$$ = $$s \cdot \mathcal{L} \{f(t) \} - f(0)$$ (11) $$\mathcal{L} \{H_c(t)\}(s)$$ = $$\dfrac{e^{-cs}}{s} \; (s>0)$$ (12) $$\mathcal{L} \{H_c(t) \cdot f(t-c)\}(s)$$ = $$e^{-cs} \mathcal{L}\{f(t)\}(s)$$ (13) $$\mathcal{L} \{H_c(t) \cdot f(t)\}(s)$$ = $$e^{-cs} \mathcal{L}\{f(t+c)\}(s)$$ (14) $$\mathcal{L} \{ \delta_c(t)\}(s)$$ = $$e^{-cs}$$ (15) $$\mathcal{L} \{e({ct} cdot f(t)\}(s)$$ = $$\mathcal{L} \{f(t)\}(s-c)$$ (16)