This, in combination with the result from the Initial Value Theorem, tells us much about
this particular signal. If this were the output of a circuit, we could now conclude that
immediatel
y after „switch on,‟ or when time is just after zero, it has a value of
Also, we
know that the response settles in the long term to a stable value of
; this is the steady
state response. We will see the full importance of these conclusions when we start to
examine circuit initial and steadystate behavior in more detail.
As with the table of common Laplace Transforms, we can also summarize the above
theorems into a quick reference table. It will be available on any test or exam that requires
you to directly calculate a Laplace Transform. Note that the last entry was not discussed
above; we will cover this later for transfer functions.
Required Reading:
Table of Laplace Properties
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27
Table of Laplace Transform Properties and Theorems
Property or Theorem
Formula or Relation
Linearity
{
}
Differentiation
2
3
Integration
2∫
3
Time Shift
{ }
Frequency Shift
{
}
Time Scaling
{
}
(
)
(Assuming
)
Initial Value Theorem
(Assuming
has no impulses)
Final Value Theorem
(Assuming
has no RHP poles)
Real Convolution
{ }
1. Calculate
{
}
2. Calculate
{
}
3. Calculate
{
}
4. Calculate
{
}
5. Calculate
,

, where
6. Calculate
,∫

, where
Practice Problems:
Laplace Transform