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Mathematics LibreTexts

Elementary Laplace Transforms

\( \mathcal{L} \{1\}(s)\) = \( \dfrac{1}{s} \; (s>0)\) (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)