1 The Laplace transform
If is a causal function then the Laplace transform of is written and defined by:
Clearly, once the integral is performed and the limits substituted the resulting expression will involve the parameter alone since the dependence upon is removed in the integration process. This resulting expression in is denoted by ; its precise form is dependent upon the form taken by . We now refine Key Point 1 (page 4).
To begin, we determine the Laplace transform of some simple causal functions. For example, if we consider the ramp function with graph
Figure 11
we find:
Now we have the difficulty of substituting in the limits of integration. The only problem arises with the upper limit ( ). We shall always assume that the parameter is so chosen that no contribution ever arises from the upper limit ( ). In this particular case we need only demand that is real and positive. Using this ‘rule of thumb’:
Thus, if then .
A similar, but more tedious, calculation yields the result that if in which is a positive integer then:
[We remember ]
Task!
Find the Laplace transform of the step function .
Begin by obtaining the Laplace integral:
You should obtain since in the range of integration, and so leading to
Now complete the integration:
You should have obtained:
where, again, we have assumed the contribution from the upper limit is zero.
As a second example, we consider the decaying exponential where is a positive constant. This function has graph:
Figure 12
In this case,
Therefore, if then .
Following this approach we can develop a table of Laplace transforms which records, for each causal function listed, its corresponding transform function . Table 1 gives a limited table of transforms.
Rule | Causal function | Laplace transform |
1 | ||
2 | ||
3 | ||
4 | ||
5 | ||
6 | ||
7 | ||
8 | ||
Note: For convenience, this table is repeated at the end of the Workbook.
1.1 The linearity property of the Laplace transformation
If and are causal functions and , are constants then
That is, the Laplace transform of a linear sum of causal functions is a linear sum of Laplace transforms. For example,
Task!
Obtain the Laplace transform of the hyperbolic function .
Begin by expressing in terms of exponential functions:
Now use the linearity property (Key Point 4) to obtain the Laplace transform of the causal function :
You should obtain since
Task!
Obtain the Laplace transform of the hyperbolic function .
You should obtain since
Task!
Find the Laplace transform of the delayed step-function , .
Write the delayed step-function here in terms of an integral:
You should obtain (note the lower limit is ) since:
In the first integral and so , therefore .
In the second integral and so , therefore . Hence
Now complete the integration:
Exercise
Determine the Laplace transform of the following functions.