For arbitrary constants and it follows that
For an arbitrary constant
For an arbitrary constant
For an arbitrary constant
For a causal function of time, , for which the derivative for exists, then as shown in section A.1.3, one obtains
This theorem shows that a differentiation of the mapped function corresponds to a multiplication with the time in the time domain:
The integral of a function is mapped by
The convolution of two functions of time and , presented by the symbolic notation , is defined as
In section A.1.3 it is shown that the convolution of the two original functions corresponds to the multiplication of the related mapped functions, that is
Whereas in h) the convolution of two functions of time was given, a similar result for the convolution of two functions in the frequency domain exists and is given by
Here and is valid. Furthermore, is the complex variable of integration. According to this theorem the Laplace transform of the product of two functions of time is equal to the convolution of and in the mapped domain. This is shown in detail in section A.1.3.
The theorem of the initial condition allows the direct calculation of the function value of a causal function of time from the Laplace transform . If the Laplace transform of and exist, then
Using the theorem of the final value the value of for can be determined from , if the Laplace transform of and exist and the limit also exists. Then it follows from section A.1.3 that
One has to observe that
or |
It can be concluded from the last two examples that the following general statement is valid: If the Laplace transform has, apart from a single pole at the origin , poles on the imaginary axis or in the right-half plane, then the initial or final value theorems cannot be applied.