Kunnskapsgnist
Laplace transformen tar en tidsavhengig funksjon, $f(t)$, og gjør den om til en frekvensavhengig funksjon, $F(s)$:
$$F(s) = \mathcal{L}(f(t)) = \int_0^{\infty} e^{-st} f(t) \; dt$$Selv om vi kan bruke formelen for å finne Laplace transformen, er det ofte raskere og enklere å bruke tabeller:
| Nr. | $f(\textcolor{blue}{t})= \mathcal{L}^{-1}(F(\textcolor{red}{s}))$ | $F(\textcolor{red}{s}) = \mathcal{L}(f(\textcolor{blue}{t}))$ | Gjelder | Utregning |
|---|---|---|---|---|
| 1 | $1$ | $\frac{1}{\textcolor{red}{s}}$ | $\textcolor{red}{s} > 0$ | årsak |
| 2 | $\textcolor{blue}{t}$ | $\frac{1}{\textcolor{red}{s}^2}$ | $\textcolor{red}{s} > 0$ | årsak |
| 3 | $\textcolor{blue}{t}^n$ | $\frac{n!}{\textcolor{red}{s}^{n+1}}$ | $\textcolor{red}{s} > 0, \quad n = \{1,2,3,\cdots\}$ | |
| 4 | $e^{a\textcolor{blue}{t}}$ | $\frac{1}{\textcolor{red}{s}-a}$ | $\textcolor{red}{s} > a$ | årsak |
| 5 | $\textcolor{blue}{t} e^{a\textcolor{blue}{t}}$ | $\frac{1}{(\textcolor{red}{s}-a)^2}$ | $\textcolor{red}{s} > a$ | |
| 6 | $\textcolor{blue}{t}^n e^{a\textcolor{blue}{t}}$ | $\frac{n!}{(\textcolor{red}{s}-a)^{n+1}}$ | $\textcolor{red}{s} > a, \quad n = \{1,2,3,\cdots\}$ | |
| 7 | $\sin(b \textcolor{blue}{t})$ | $\frac{b}{\textcolor{red}{s}^2 + b^2}$ | $\textcolor{red}{s} > 0$ | årsak |
| 8 | $\cos(b \textcolor{blue}{t})$ | $\frac{\textcolor{red}{s}}{\textcolor{red}{s}^2 + b^2}$ | $\textcolor{red}{s} > 0$ | årsak |
| 9 | $e^{a\textcolor{blue}{t}}\sin(b \textcolor{blue}{t})$ | $\frac{b}{(\textcolor{red}{s}-a)^2 + b^2}$ | $\textcolor{red}{s} > a$ | |
| 10 | $e^{a\textcolor{blue}{t}}\cos(b \textcolor{blue}{t})$ | $\frac{\textcolor{red}{s}-a}{(\textcolor{red}{s}-a)^2 + b^2}$ | $\textcolor{red}{s} > a$ | |
| 11 | $u(\textcolor{blue}{t}-a)$ | $\frac{1}{\textcolor{red}{s}} e^{-a\textcolor{red}{s}}$ | $\textcolor{red}{s} > 0, \quad a \ge 0$ | årsak |
| 12 | $\delta(\textcolor{blue}{t})$ | $1$ | $\textcolor{red}{s} > 0$ | årsak |
| 13 | $\delta(\textcolor{blue}{t-a})$ | $e^{-a\textcolor{red}{s}}$ | $\textcolor{red}{s} > 0, \quad a \ge 0$ | årsak |
| 14 | $g(\textcolor{blue}{t}) + h(\textcolor{blue}{t})$ | $\mathcal{L}(g(\textcolor{blue}{t})) + \mathcal{L}(h(\textcolor{blue}{t}))$ | Laplace transformasjonen er lineær | |
| 15 | $ag(\textcolor{blue}{t})$ | $a\mathcal{L}(g(\textcolor{blue}{t}))$ | $a$ er konstant | Laplace transformasjonen er lineær |
| 16 | $h(t)u(t-a)$ | $e^{-as}\mathcal{L}(h(t+a))$ | $a > 0$ | årsak |
| 17 | $h(t-a)u(t-a)$ | $e^{-as}\mathcal{L}(h(t))$ | $a > 0$ | årsak |
| 18 | $f'(t)$ | $sF(s) - f(0)$ | derivasjon | årsak |
| 19 | $f''(t)$ | $s^2F(s) - sf(0) - f'(0)$ | derivasjon | årsak |
| 20 | $\int_0^t g(\tau) \: d \tau$ | $\frac{1}{s}G(s)$ | integrasjon | årsak |
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