**The Laplace Transform**

Part 4: Powers

Let us now find the Laplace transform for powers of *t*; letting *f*(*t*)=*t*^{r}, 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 ,

.

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Tags: Gamma Function, Laplace Transform, Math, Monday Math

This entry was posted on January 11, 2010 at 12:12 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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