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Direct verification of a Laplace and Fourier transform pair

Verify that f(t)=t^n, F(s)=n!/s^(n+1) are a Laplace transform pair.

Verify that f(x)=1 if |x|<a and f(x)=0 otherwise and F(y)=(2/y)sin(ay) are a Fourier transform pair.

The question is repeated with correct mathematical notation in the attachment (question 5).


Solution Summary

When applying Laplace and Fourier transforms and inverse transforms, we usually use tables of known results and various shifting theorems. Constructing these tables involves calculations directly from the integral definitions. This can be a somewhat tricky process, especially calculating inverse transforms directly (although the theory enables us to avoid having to do this). The solution is a 2 page Word attachment verifying a particular Laplace transform pair, and a particular Fourier transform pair directly from the integral definitions.