Let Γ(1 + x) = √(2πx)x^x e^–x φ(x);This result is Stirling's theorem. A simple proof is given in § 1.87 of Titchmarsh's Theory of Functions (Oxford Univ. Press, 1932).Rather more than Stirling's theorem can be proved by a method which assumes nothing but the definition of the Γ-function, and Γ (½) = √π, from which it follows that