The discussion focuses on expanding the gamma function, specifically Γ[(1/2) ± (ε/2)], around the pole ε at first order in ε. The key findings include the approximations Γ(½ ± ε/2) ≈ Γ(½) ± ε/2 Γ'(½) and the use of the digamma function, ψ(x) = Γ'(x)/Γ(x). Important values such as Γ(½) = √π and ψ(½) = -γ - 2 ln 2 are highlighted, leading to the expansions: Γ(½ - ε/2) = √π + (1/2)√π ε(γ_E + log(4)) + O(ε²) and Γ(½ + ε/2) = √π + (√π ε ψ(½))/2 + O(ε²).
PREREQUISITESMathematicians, physicists, and students studying complex analysis or special functions, particularly those interested in the properties and expansions of the gamma function.
