## Monday Math 103

The Laplace Transform
Part 4: Powers

Let us now find the Laplace transform for powers of t; letting f(t)=tr, we use the definition of the Laplace transform to find:
,
with  as the region of convergence.
Using the u-substitution u=st,
.
You should recognise this last integral as the gamma function of r+1, and so
,
and so we see that it is necessary that r not be a negative integer. If r is a non-negative integer n, then we have
.
Note also that r=0 gives a transform of 1/s, which is what we previously found for the Laplace transform of a constant.
Setting , and using , we see
.
Similarly, since ,
.

### 3 Responses to “Monday Math 103”

1. Monday Math 105 « Twisted One 151's Weblog Says:

[…] in the s domain. This can be combined with our results for sine and cosine to get . And our results for powers of t to get . Note that for all of these, the shift in the “frequency” […]

2. Monday Math 107 « Twisted One 151's Weblog Says:

[…] let us try to find the function f(t) with Laplace transform . Using , we see . Now, we recall from here and here that and , so we see that , and so we see that f(t) is times the convolution of and , […]

3. Monday Math 110 « Twisted One 151's Weblog Says:

[…] the ramp function, . From here, we know , and so the above tells us that , as expected from here. In fact, we can apply the formula repeatedly , and in general, ; compare to our earlier results […]