{ f(t)} =
Time Function f(t) 
Laplace Transform of f(t) 
F1 
s > 0 
t (unitramp function) 
s > 0 
t^{n} (n, a positive integer) 
s > 0 
e^{at} 
s > a 
sin ωt 
s > 0 
cos ωt 
s > 0 
t^{n}g(t), for n = 1, 2, ... 

t sin ωt 
s > ω 
t cos ωt 
s > ω 
g(at) 
Scale 
e^{at}g(t) 
G(s − a) Shift property 
e^{at}t^{n}, for n = 1, 2, ... 
s > a 
te^{t} 
s > 1 
1 − e^{}^{t}^{/}^{T} 
s > 1/T 
e^{at}sin ωt 
s > a 
e^{at}cos ωt 
s > a 
u(t) 
s > 0 
u(t − a) 
s > 0 
u(t − a)g(t − a) 
e^{}^{as}G(s) 
g'(t) 
sG(s) − g(0) 
g''(t) 
s^{2} • G(s) − s • g(0) − g'(0) 
g^{(}^{n}^{)}(t) 
s^{n} • G(s) − s^{n}^{1} • g(0) − s^{n}^{2} • g'(0) − ... − g^{(}^{n}^{1)}(0) 



· Property1  Constant Multiple
· If
a is a constant and f(t) is a function of t, then L{a f(t)} = a L{f(t)}
· Example
{7 sin t} = 7{sin t}
· Property2  Linearity Property
· If
a and b are constants while f(t) and g(t) are functions of t, then
L{a f(t) + b g(t)} = a L{f(t)}
+ b L{g(t)}
· Example
L{3t + 6t2 } = 3 L{t} + 6 L{t2}
· Property3. Change of Scale Property
· If
L{f(t)} = F(s) then L{f(at)} = 1/a F(s/a)
· Example
L{F(5t) = (1/5)F(s/5)
· Property4. Shifting Property (Shift Theorem)
· L{exp(at)f(t)}
= F(s − a)
· Example
L{exp(3t)f(t)} = F(s − 3)
· Property5. Differential transformation
· L{tf(t)}
= F’(s)
· Property6.
· The Laplace transforms of the real (or imaginary) part of a complex function is
equal to the real (or imaginary) part of the transform of the complex function.
· Let Re denote the real part of a complex function C(t) and Im denote the imaginary
part of C(t), then L{Re[C(t)]} = Re L{C(t)} and L{Im[C(t)]} = Im L{C(t)}