1 The second shift theorem
The second shift theorem is similar to the first except that, in this case, it is the time-variable that is shifted not the -variable. Consider a causal function which is shifted to the right by amount , that is, the function where . Figure 13 illustrates the two causal functions.
Figure 13
The Laplace transform of the shifted function is easily obtained:
(Note the change in the lower limit from to resulting from the step function switching on at ). We can re-organise this integral by making the substitution . Then
Therefore
The final integral is simply the Laplace transform of , which we know is and so, finally, we have the statement of the second shift theorem:
Obviously, this theorem has its uses in finding the Laplace transform of time-shifted causal functions but it is also of considerable use in finding inverse Laplace transforms since, using the inverse formulation of the theorem of Key Point 8 we get:
Task!
Find the inverse Laplace transform of .
You should obtain for the following reasons. We know that the inverse Laplace transform of is (Table 1, Rule 3) and so, using the second shift theorem (with ), we have
This function is graphed in the following figure:
Task!
Find the inverse Laplace transform of
You should obtain .
To obtain this, complete the square in the denominator: and so
Now, using the first shift theorem
and
Thus