オイラーの定数
Ψ ( z ) = d d z log  Γ ( z ) = Γ ′ ( z ) Γ ( z ) = lim n → ∞ ( log  n − ∑ k = 0 n 1 z + k ) {\displaystyle {\begin{aligned}\Psi (z)&={\frac {d}{dz}}\log \Gamma (z)={\frac {\Gamma '(z)}{\Gamma (z)}}\\&=\lim _{n\to \infty }\left(\log {n}-\sum _{k=0}^{n}{\frac {1}{z+k}}\right)\end{aligned}}}
に z = 1 {\displaystyle z=1} を代入すると
Ψ ( 1 ) = Γ ′ ( 1 ) = lim n → ∞ ( log  n − ∑ k = 0 n 1 1 + k ) = lim n → ∞ ( − γ − 1 1 + n ) = − γ {\displaystyle {\begin{aligned}\Psi (1)=\Gamma '(1)&=\lim _{n\to \infty }\left(\log {n}-\sum _{k=0}^{n}{\frac {1}{1+k}}\right)\\&=\lim _{n\to \infty }\left(-\gamma -{\frac {1}{1+n}}\right)\\&=-\gamma \\\end{aligned}}}
を得る。