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Consider the Laplace transform of t p, where p > ?1. (a)

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Consider the Laplace transform of t p, where p > ?1. (a) Referring to Problem 36, show that ?{t p } = ? ? 0 e?stt p dt =

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Consider the Laplace transform of t p, where p > ?1. (a) Referring to Problem 36, show that ?{t p } = ? ? 0 e?stt p dt = 1 sp+1 ? ? 0 e?x xp dx = ?(p + 1) sp+1 , s > 0. (b) Let p be a positive integer n in (a). Show that ?{t n } = n! sn+1 , s > 0. (c) Show that ?{t ?1?2 } = 2 ?s ? ? 0 e?x2 dx, s > 0. It is possible to show that ? ? 0 e?x2 dx = ?? 2 ; hence, ?{t ?1?2 } = ?? s , s > 0. (d) Show that ?{t 1?2 } = ?? 2s3?2 , s > 0.

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