The Laplace transform, named after the French mathematician Pierre-Simon de Laplace (1749-1827), of a function f(x) is given by L{f(t)}(s) =

Laplace transforms are useful for solving differential equations.
(a) Show that the Laplace transform of ta is given by Γ(a + 1) / sa+1 and is defined for s > 0.
(b) Show that the Laplace transform of eat is given by 1/(s - a) and is defined for s > a. (c) Show that the Laplace transform of sin(at) is given by a/(s2 + a2) and is defined for s > 0?

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