Quote:
Originally Posted by retina
Presumably n! = Γ(n + 1). For n >= 0 that would suffice, right? What did I miss?

You are missing that n! = Γ(n+1) + sin(πn). Or Γ(n+1) + sin(π(n+1)); take your pick... Wikipedia is helpful
to demonstrate that both are valid analytic continuations of the factorials to the nonintegers.
Do Γ(n+1) and Γ(n+1) + sin(πn) have the same derivatives?