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) |
