11.2: Common Laplace Transforms
- Page ID
- 22908
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| Signal | Laplace Transform | Region of Convergence |
|---|---|---|
| \(\delta(t)\) | \(1\) | All \(s\) |
| \(\delta(t-T)\) | \(e^{(-sT)}\) | All \(s\) |
| \(u(t)\) | \(\frac{1}{s}\) | \(\operatorname{Re}(s)>0\) |
| \(u(-t)\) | \(\frac{1}{s}\) | \(\operatorname{Re}(s)<0\) |
| \(tu(t)\) | \(\frac{1}{s^2}\) | \(\operatorname{Re}(s)>0\) |
| \(t^{n} u(t)\) | \(\frac{n !}{s^{n+1}}\) | \(\operatorname{Re}(s)>0\) |
| \(-(t^n u(-t))\) | \(\frac{n !}{s^{n+1}}\) | \(\operatorname{Re}(s)<0\) |
| \(e^{-(\lambda t)} u(t)\) | \(\frac{1}{s+\lambda}\) | \(\operatorname{Re}(s)>-\lambda\) |
| \(\left(-e^{-(\lambda t)}\right) u(-t)\) | \(\frac{1}{s+\lambda}\) | \(\operatorname{Re}(s)<-\lambda\) |
| \(t e^{-(\lambda t)} u(t)\) | \(\frac{1}{(s+\lambda)^{2}}\) | \(\operatorname{Re}(s)>-\lambda\) |
| \(t^{n} e^{-(\lambda t)} u(t)\) | \(\frac{n !}{(s+\lambda)^{n+1}}\) | \(\operatorname{Re}(s)>-\lambda\) |
| \(-\left(t^{n} e^{-(\lambda t)} u(-t)\right)\) | \(\frac{n !}{(s+\lambda)^{n+1}}\) | \(\operatorname{Re}(s)<-\lambda\) |
| \(\cos (b t) u(t)\) | \(\frac{s}{s^{2}+b^{2}}\) | \(\operatorname{Re}(s)>0\) |
| \(\sin (b t) u(t)\) | \(\frac{b}{s^{2}+b^{2}}\) | \(\operatorname{Re}(s)>0\) |
| \(e^{-(a t)} \cos (b t) u(t)\) | \(\frac{s+a}{(s+a)^{2}+b^{2}}\) | \(\operatorname{Re}(s)>-a\) |
| \(e^{-(a t)} \sin (b t) u(t)\) | \(\frac{b}{(s+a)^{2}+b^{2}}\) | \(\operatorname{Re}(s)>-a\) |
| \(\frac{d^{n}}{d t^{n}} \delta(t)\) | \(s^n\) | All \(s\) |