The gamma function, which plays an important role in advanced applications, is defined for n ≥1 by Γ^'(n)=∫0^∞ t^n-1 e^-t d t (a) Show that the integral defining Γ(n) converges for n ≥1 (it actually converges for all n>0 ) . Hint: Show that t^n-1 e^-t<t^-2 for t sufficiently large. (b) Show that Γ^'(n 1)=n Γ(n) using Integration by Parts. (c) Show that Γ^'(n 1)=n ! if n ≥1 is an integer. Hint: Use (a) repeatedly. Thus, Γ^'(n) provides a way of defining n -factorial when n is not an integer. 100

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