EDB — 1BM

[1BM]We define the Gamma function \(Γ:(0,∞)→ℝ\) as

• Show that \(Γ(x)\) is well defined for \(x{\gt}0\) real.

• Show that \(Γ(x+1)=x Γ(x)\) and deduce that \(Γ(n+1)=n!\) for \(n∈ℕ\).

• Show that \(Γ(x)\) is analytic.

  (You can assume that derivatives of \(Γ\) are \(Γ^{(n)}(x) = ∫_ 0^∞ (\log t)^ n t^{x-1}e^{-t}\, {\mathbb {d}}t\); those are obtained by derivation under integral sign.)