The properties associated with the Laplace transform are similar to those of the Fourier transform. First, let's set define some notation, we will use the notation
With this notation defined, lets now look at some properties.
Linearity
Given that
The linearity property follows easily using the definition of the Laplace transform.
Time Delay
The reason we call this the time delay property rather than the time shift property is that the time shift must be positive, i.e. if
Letting
where we note that the first integral in the last line is zero since
sShift
This property is the Laplace transform corresponds to the frequency shift property of the Fourier transform. In fact, the derivation of the
The
Multiplication by t t
Let's begin by taking the derivative of the Laplace transform:
So we can write
This idea can be extended to multiplication by
Proceeding in this manner, we find that
Time Scaling
The time scaling property for the Laplace transform is similar to that of the Fourier transform:
where in the second equality, we made the substitution
Convolution
The derivation of the convolution property for the Laplace transform is virtually identical to that of the Fourier transform. We begin with
Applying the timedelay property of the Laplace transform gives
If
Differentiation
The Laplace transform of the derivative of a signal will be used widely. Consider
this can be integrated by parts:
which gives
therefore we have,
Higher Order Derivatives
The previous derivation can be extended to higher order derivatives. Consider
it follows that
which leads to
This process can be iterated to get the Laplace transform of arbitrary higher order derivatives, giving
where it should be understood that
Integration
Let
it follows that
and
therefore
but since
we have
Now suppose
The quantity
it follows that
where we have used the fact that
The Initial Value Theorem
The initial value theorem makes it possible to determine
Taking the limit
There are two cases, the first is when
Since
The second case is when
For example, if we integrate the righthand side of Equation 36 with
The lefthand side of Equation 37 can be written as
From the sifting property of the unit impulse, the first term in Equation 38 is
while the second term is zero since in the limit, the real part of
The Final Value Theorem
The Final Value Theorem allows us to determine
from
The lefthandside of Equation 41 can be written as
Substituting this result back into Equation 41 leads to the Final Value Theorem
which is only valid as long as the limit
Property 


Linearity 


Time Delay 


sShift 


Multiplication by 


Multiplication by 


Convolution 


Differentiation 








Integration 


Initial Value Theorem 


Final Value Theorem 
