This in combination with the result from the Initial Value Theorem tells us

This in combination with the result from the initial

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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 steady-state 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
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