Properties of the Laplace TransformFunction f(t) Laplace Transform F(s) Commentc 1 f(t) c 1 F(s) linearity property
f′(t) sF (s)−f(0 +) Derivative property
f′′(t) s^2 F(s)−sf (0 +)−f′(0+) Derivative property
f(n)(t)snF(s)−sn−^1 f(0 +)−
···− sf (n−2)(0 +)−f(n−1)(0 +)Derivative property
∫t
0 f(τ)dτ1sF(s) s >^0 Integration transform
c 1 f(t) + c 2 g(t) c 1 F(s) + c 2 G(s) linearity property
tf (t) −F′(s) multiplication by t property
t^2 f(t) (−1)^2 F′′(s)tnf(t) (−1)nF(n)(s)eαtf(t) F(s−α) First shift property
1
tf(t)∫∞s F(s)ds Division by tproperty
f(t−α)H(t−α) e−αsF(s) Second shift property
1
αf(tα) F(αs) scaling property
1
αeβt/α f(tα) f(αs −β) shifting scaling
f(t+p) = f(t) 1 −e^1 −ps∫p0 e−stf(t)dt periodic property
∫t