オイラーの定数
γ = lim m → ( − 1 ) / ( k + 1 ) ( ∑ n = 2 ∞ ( − 1 ) n ζ ( n ) m n − 1 n − m ln  Γ ( m + 1 m ) ) , { k ∣ k ∈ Z ∖ { − 1 } } {\displaystyle \gamma =\lim _{m\rightarrow (-1)/(k+1)}\left(\displaystyle \sum _{n=2}^{\infty }{\frac {(-1)^{n}\zeta (n)}{m^{n-1}n}}-m\ln \Gamma \left({\frac {m+1}{m}}\right)\right),\quad \{\,k\,\mid \,k\in \mathbb {Z} \setminus \{-1\}\}}
γ = 3 2 − ln  2 − ∑ n = 2 ∞ ( − 1 ) n ( n − 1 ) ( ζ ( n ) − 1 ) n {\displaystyle \gamma ={\frac {3}{2}}-\ln 2-\sum _{n=2}^{\infty }{\frac {(-1)^{n}(n-1)(\zeta (n)-1)}{n}}}
γ = 1 − ln  2 + ∑ n = 2 ∞ ( − 1 ) n ( ζ ( n ) − 1 ) n {\displaystyle \gamma =1-\ln 2+\sum _{n=2}^{\infty }{\frac {(-1)^{n}(\zeta (n)-1)}{n}}}
等々.
オイラーの定数の値は以下の定積分で与えられる。
γ = − Γ ′ ( 1 ) = − ∫ 0 ∞ log  t e − t d t = − ∫ 0 1 log  log  1 u d u ( u = e − t ) = − ∫ − ∞ ∞ u e u − e u d u ( u = log  t ) {\displaystyle {\begin{aligned}\gamma &=-\Gamma '(1)\\&=-\int _{0}^{\infty }\log {t}e^{-t}dt\\&=-\int _{0}^{1}\log \log {\frac {1}{u}}du\qquad (u=e^{-t})\\&=-\int _{-\infty }^{\infty }ue^{u-e^{u}}du\qquad (u=\log {t})\\\end{aligned}}}
あるいは
log  t = ∫ 1 t 1 s d s = ∫ 1 t ∫ 0 ∞ e − s u d u d s = ∫ 0 ∞ ∫ 1 t e − s u d s d u = ∫ 0 ∞ e − u − e − t u u d u {\displaystyle {\begin{aligned}\log {t}&=\int _{1}^{t}{\frac {1}{s}}ds\\&=\int _{1}^{t}\int _{0}^{\infty }e^{-su}duds\\&=\int _{0}^{\infty }\int _{1}^{t}e^{-su}dsdu\\&=\int _{0}^{\infty }{\frac {e^{-u}-e^{-tu}}{u}}du\\\end{aligned}}}