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