Time Function f(t) 
  f(t) = laplace-1{F(s)} 		Laplace Transform of f(t)
F(s) = laplace{ f(t)}
1 1 on ss > 0
t (unit-ramp function) 1 on s^2s > 0
tn (n, a positive integer) n1 on s^(n+1)s > 0
eat 1 on s - as > a
sin ωt omega on s^2 + omega^2s > 0
cos ωt s on s^2 + omega^2s > 0
tng(t), for n = 1, 2, ... math
t sin ωt 2wss > |ω|
t cos ωt ((s²-ω²)/((s²+ω²)²))s > |ω|
g(at) (1/a)G((s/a))   Scale property
eatg(t) G(s − a)   Shift property
eattn, for n = 1, 2, ... ((n!)/((s-a)ⁿ+¹)) s > a
te-t (1/((s+1)²))s > -1
1 − e-t/T (1/(s(1+Ts)))s > -1/T
eatsin ωt (ω/((s-a)²+ω²))s > a
eatcos ωt we s > a
u(t) 1 on ss > 0
u(t − a) ((e_{}^{-as})/s)s > 0
u(t − a)g(t − a) e-asG(s)   Time-displacement theorem
g'(t) sG(s) − g(0)
g''(t) s2 • G(s) − s • g(0) − g'(0)
g(n)(t) sn • G(s) − sn-1 • g(0) − sn-2 • g'(0) − ... − g(n-1)(0)
∫₀^{t}g(t)dt ((G(s))/s)
∫g(t)dt ((G(s))/s)+(1/s){∫g(t)dt}_{t=0}