Lesson 9 · The Gamma function — meromorphic continuation, poles, and residues
Step the recursion to watch the domain of Γ grow one strip at a time, and a pole appear at the price of each strip.
Re(s) > 0: the integral converges here and nowhere else.
Γ(s) = ∫₀^∞ t^(s−1) e^(−t) dt
Domain: Re(s) > 0
Algebra supplies existence and agreement; the identity theorem supplies uniqueness.
Drag the point on the plane to read Γ(s) there.