 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-2-course-of-the-university-of-oxford/2007/9/ii/solution

For

Re p > 0

, the starting integral is

\int_{0}^{\infty} e^{- pt} d t = 1/ p

. Repeated integration by parts, with vanishing boundary terms, gives

L [t^{n}] (p) = \frac{n}{p} L [t^{n - 1}] (p) = \frac{n !}{p ^{n + 1}} (n \geq 0) .

This also follows from the gamma function. The stated open half-plane ensures absolute convergence.

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