Electronics/Laplace Transform pairs

 ${\displaystyle f(t)}$ ${\displaystyle F(s)}$ ${\displaystyle \delta (t-t_{0})}$ ${\displaystyle e^{-st_{0}}}$ ${\displaystyle u(t)}$ ${\displaystyle {\frac {1}{s}}}$ ${\displaystyle e^{-\alpha t}u(t)}$ ${\displaystyle {\frac {1}{s+\alpha }}}$ ${\displaystyle \sin(\beta t)u(t)}$ ${\displaystyle {\frac {\beta }{s^{2}+\beta ^{2}}}}$ ${\displaystyle \cos(\beta t)u(t)}$ ${\displaystyle {\frac {s}{s^{2}+\beta ^{2}}}}$ ${\displaystyle e^{-\alpha t}\sin(\beta t)u(t)}$ ${\displaystyle {\frac {\beta }{(s+\alpha )^{2}+\beta ^{2}}}}$ ${\displaystyle e^{-\alpha t}\cos(\beta t)u(t)}$ ${\displaystyle {\frac {s+\alpha }{(s+\alpha )^{2}+\beta ^{2}}}}$ ${\displaystyle 2e^{\alpha t}(a\cos {\beta t}-b\sin {\beta t})u(t)}$ ${\displaystyle {\frac {a+jb}{s+\alpha -j\beta }}+{\frac {a-jb}{s+\alpha +j\beta }}}$ ${\displaystyle t^{n}}$ ${\displaystyle {\frac {n!}{s^{n+1}}}}$ ${\displaystyle t^{n}f(t)}$ ${\displaystyle (-1)^{n}{\frac {d^{n}F(s)}{ds^{n}}}}$ ${\displaystyle {\frac {df(t)}{dt}}}$ ${\displaystyle sF(s)-f(0)}$ ${\displaystyle \int _{0}^{t}f(x)dx}$ ${\displaystyle {\frac {1}{s}}F(s)}$ ${\displaystyle u_{-1}(t-t_{0})f(t-t_{0})}$ ${\displaystyle e^{-t_{0}s}F(s),t_{0}\geq 0}$ ${\displaystyle h(t)*f(t)=\int _{0}^{t}h(t-\tau )f(\tau )d\tau }$ ${\displaystyle H(s)F(s)}$ ${\displaystyle f_{1}(t)f_{2}(t)}$ ${\displaystyle F_{1}(s)*F_{2}(s)}$