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11.2: Common Laplace Transforms

Table $$\PageIndex{1}$$
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 $$s$$