The function $$u$$ is the Heaviside function, $$\delta$$ is the Dirac delta function, and
 $$f(t)$$ $$F(s) = \mathcal{L} \bigl\{ f(t) \bigr\}= \int_0^\infty e^{-st} f(t) \, dt$$ $$C$$ $$\frac{C}{s}$$ $$t$$ $$\frac{1}{s^2}$$ $$t^2$$ $$\frac{2}{s^3}$$ $$t^n$$ $$\frac{n!}{s^{n+1}}$$ $$t^p \quad (p > 0)$$ $$\frac{\Gamma(p+1)}{s^{p+1}}$$ $$e^{-at}$$ $$\frac{1}{s+a}$$ $$\sin (\omega t)$$ $$\frac{\omega}{s^2+\omega^2}$$ $$\cos (\omega t)$$ $$\frac{s}{s^2+\omega^2}$$ $$\sinh (\omega t)$$ $$\frac{\omega}{s^2-\omega^2}$$ $$\cosh (\omega t)$$ $$\frac{s}{s^2-\omega^2}$$ $$u(t-a)$$ $$\frac{e^{-as}}{s}$$ $$\delta(t)$$ $$1$$ $$\delta(t-a)$$ $$e^{-as}$$ $$\operatorname{erf}\left( \frac{t}{2a} \right)$$ $$\frac{1}{s} e^{(as)^2} \operatorname{erfc}(as)$$ $$\frac{1}{\sqrt{\pi t}} \exp\left(\frac{-a^2}{4t}\right) \quad (a \geq 0)$$ $$\frac{e^{-as}}{\sqrt{s}}$$ $$\frac{1}{\sqrt{\pi t}} - a e^{a^2 t} \operatorname{erfc}(a \sqrt{t}) \quad (a>0)$$ $$\frac{1}{\sqrt{s}+a}$$ $$a f(t) + b g(t)$$ $$a F(s) + bG(s)$$ $$f(at) \quad (a > 0)$$ $$\frac{1}{a}F\left( \frac{s}{a} \right)$$ $$f(t-a)u(t-a)$$ $$e^{-as} F(s)$$ $$e^{-at} f(t)$$ $$F(s+a)$$ $$g'(t)$$ $$sG(s)-g(0)$$ $$g''(t)$$ $$s^2G(s)-sg(0)-g'(0)$$ $$g'''(t)$$ $$s^3G(s)-s^2g(0)-sg'(0)-g''(0)$$ $$g^{(n)}(t)$$ $$s^nG(s)-s^{n-1}g(0)-\cdots-g^{(n-1)}(0)$$ $$(f * g)(t) = \int_0^t f(\tau) g(t-\tau) \, d\tau$$ $$F(s)G(s)$$ $$tf(t)$$ $$-F'(s)$$ $$t^nf(t)$$ $${(-1)}^nF^{(n)}(s)$$ $$\int_0^t f(\tau) d\tau$$ $$\frac{1}{s} F(s)$$ $$\frac{f(t)}{t}$$ $$\int_s^\infty F(\sigma) d\sigma$$