14.6: A Function Times tⁿ
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I'll just give this one with out proof:
For n a positive integer,
L(tny)=(−1)nd2ˉydsn.
Before proceeding further, I strongly recommend that you now apply theorems 14.3.1, 14.4.1, 14.5.1 and 14.6.1 to the several entries in your existing table of Laplace transforms and greatly expand your table of Laplace transforms. For example, you can already add (sinat)/t, te−at and t2e−t to the list of functions for which you have calculated the Laplace transforms.