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Mathematics LibreTexts

Elementary Laplace Transforms

( \newcommand{\kernel}{\mathrm{null}\,}\)

\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)



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