Sharp Smith’s bounds for the gamma function

摘要

Among various approximation formulas for the gamma function, Smith showed that Γ(x+12)∼S(x)=2π(xe)x(2xtanh12x)x/2,x→∞, which is a little-known but accurate and simple one. In this note, we prove that the function x ↦ lnΓ(x + 1/2)−lnS(x) is strictly increasing and concave on (0, ∞), which shows that Smith’s approximation is just an upper one.