structural:reformatting, quality:correction
Raabe's formula

In 1840 Joseph Ludwig Raabe proved that ∫ a a + 1 l o g Γ (z) d z = 1 2 log 2 π + a log a − a, a > 0. {\displaystyle \int _{a}^{a+1}\operatorname {log\Gamma } (z)\,dz={\tfrac {1}{2}}\log 2\pi +a\log a-a,\quad a>0.} In particular, if a = 0 {\displaystyle a=0} then ∫ 0 1 l o g Γ (z) d z = 1 2 log 2 π. {\displaystyle \int _{0}^{1}\operatorname {log\Gamma } (z)\,dz={\tfrac {1}{2}}\log 2\pi.}

The latter can be derived taking the logarithm in the above multiplication formula, which gives an expression for the Riemann sum of the integrand. Taking the limit for a → ∞ {\displaystyle a\to \infty } gives the formula.