13.6: Table of Laplace transforms
- Page ID
- 51240
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We assume that \(f(t) = 0\) for \(t < 0\).
Function Transform
\(\begin{array} {lclcl} {f(t)} & \ \ \ \ \ \ \ \ \ & {F(s) = \int_{0}^{\infty} f(t) e^{-st} \ dt} & \ & {\text{(Definition)}} \\ {af(t) + bg(t)} & \ \ \ \ \ \ \ \ \ & {aF(s) + bG(s)} & \ & {\text{(Linearity)}} \\ {e^{at} f(t)} & \ \ \ \ \ \ \ \ \ & {F(s - a)} & \ & {(s-\text{shift})} \\ {f'(t)} & \ \ \ \ \ \ \ \ \ & {sF(s) - f(0)} & \ & {} \\ {f''(t)} & \ \ \ \ \ \ \ \ \ & {s^2 F(s) - sf(0) - f'(0)} & \ & {} \\ {f^{(n)} (t)} & \ \ \ \ \ \ \ \ \ & {s^n F(s) - s^{n - 1} f(0) - \ \cdot\cdot\cdot - f^{(n - 1)} (0)} & \ & {} \\ {tf(t)} & \ \ \ \ \ \ \ \ \ & {-F'(s)} & \ & {} \\ {t^n f(t)} & \ \ \ \ \ \ \ \ \ & {(-1)^n F^{(n)} (s)} & \ & {} \\ {f(t - a)} & \ \ \ \ \ \ \ \ \ & {e^{-as} F(s)} & \ & {(t-\text{translation or } t-\text{shift})} \\ {\int_{0}^{t} f(\tau) \ d\tau} & \ \ \ \ \ \ \ \ \ & {\dfrac{F(s)}{s}} & \ & {\text{(integration rule)}} \\ {\dfrac{f(t)}{t}} & \ \ \ \ \ \ \ \ \ & {\int_{s}^{\infty} F(\sigma)\ d\sigma} & \ & {} \end{array}\)
Function Transform Region of convergence
\(\begin{array} {lclcl} {1} & \ \ \ \ \ \ \ \ \ & {1/s} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > 0} \\ {e^{at}} & \ \ \ \ \ \ \ \ \ & {1/(s - a)} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > \text{Re} (a)} \\ {t} & \ \ \ \ \ \ \ \ \ & {1/s^2} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > 0} \\ {t^n} & \ \ \ \ \ \ \ \ \ & {n!/s^{n + 1}} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > 0} \\ {\cos (\omega t)} & \ \ \ \ \ \ \ \ \ & {s/(s^2 + \omega ^2)} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > 0} \\ {\sin (\omega t)} & \ \ \ \ \ \ \ \ \ & {\omega /(s^2 + \omega ^2)} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > 0} \\ {e^{at} \cos (\omega t)} & \ \ \ \ \ \ \ \ \ & {(s - a)/((s - a)^2 + \omega ^2)} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > \text{Re} (a)} \\ {e^{at} \sin (\omega t)} & \ \ \ \ \ \ \ \ \ & {\omega /((s - a)^2 + \omega ^2)} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > \text{Re} (a)} \\ {\delta (t)} & \ \ \ \ \ \ \ \ \ & {1} & \ \ \ \ \ \ \ \ & {\text{all } s} \\ {\delta (t - a)} & \ \ \ \ \ \ \ \ \ & {e^{-as}} & \ \ \ \ \ \ \ \ & {\text{all } s} \\ {\cosh (kt) = \dfrac{e^{kt} + e^{-kt}}{2}} & \ \ \ \ \ \ \ \ \ & {s/(s^2 - k^2)} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > k} \\ {\sinh (kt) = \dfrac{e^{kt} - e^{-kt}}{2}} & \ \ \ \ \ \ \ \ \ & {k/(s^2 - k^2)} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > k} \\ {\dfrac{1}{2\omega ^3} (\sin (\omega t) - \omega t \cos (\omega t))} & \ \ \ \ \ \ \ \ \ & {\dfrac{1}{(s^2 + \omega ^2)^2}} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > 0} \\ {\dfrac{t}{2\omega} \sin (\omega t)} & \ \ \ \ \ \ \ \ \ & {\dfrac{s}{(s^2 + \omega ^2)^2}} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > 0} \\ {\dfrac{1}{2\omega} (\sin (\omega t) + \omega t \cos (\omega t))} & \ \ \ \ \ \ \ \ \ & {\dfrac{s^2}{(s^2 + \omega ^2)^2}} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > 0} \\ {t^n e^{at}} & \ \ \ \ \ \ \ \ \ & {n!/(s - a)^{n + 1}} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > \text{Re} (a)} \\ {\dfrac{1}{\sqrt{\pi t}}} & \ \ \ \ \ \ \ \ \ & {\dfrac{1}{\sqrt{s}}} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > 0} \\ {t^a} & \ \ \ \ \ \ \ \ \ & {\dfrac{\Gamma (a + 1)}{s^{a + 1}}} & \ \ \ \ \ \ \ \ & {\text{Re} (s) > 0} \end{array}\)