Gamma function
Stirling's formula

The behavior of Γ (x) {\displaystyle \Gamma (x)} for an increasing positive real variable is given by Stirling's formula Γ (x + 1) ∼ 2 π x (x e) x, {\displaystyle \Gamma (x+1)\sim {\sqrt {2\pi x}}\left({\frac {x}{e}}\right)^{x},} where the symbol ∼ {\displaystyle \sim } means asymptotic convergence: the ratio of the two sides converges to 1 in the limit x → + ∞ {\textstyle x\to +\infty }. This growth is faster than exponential, exp (β x) {\displaystyle \exp(\beta x)}, for any fixed value of β {\displaystyle \beta }.

Another useful limit for asymptotic approximations for x → + ∞ {\displaystyle x\to +\infty } is: Γ (x + α) ∼ Γ (x) x α, α ∈ C. {\displaystyle {\Gamma (x+\alpha)}\sim {\Gamma (x)x^{\alpha }},\qquad \alpha \in \mathbb {C}.}

When writing the error term as an infinite product, Stirling's formula can be used to define the gamma function: Γ (x) = 2 π x (x e) x ∏ n = 0 ∞ e − 1 (1 + 1 x + n) 1 2 + x + n {\displaystyle \Gamma \left(x\right)={\sqrt {\frac {2\pi }{x}}}\left({\frac {x}{e}}\right)^{x}\prod _{n=0}^{\infty }e^{-1}\left(1+{\frac {1}{x+n}}\right)^{{\frac {1}{2}}+x+n}}
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