= Derivative of the gamma function
{title2=$\Gamma'(a)=\int_0^\infty e^{-t}t^{a-1}\log t\,dt$}
= Derivatives of the gamma function
{synonym}
For positive real $a$, differentiating the <Gamma function> integral under the integral sign gives the displayed formula; successive <derivatives> insert higher powers of $\log t$. Dominated differentiation on compact positive-$a$ intervals is justified by exponential decay at infinity and integrability of $t^{a-1}|\log t|^n$ at zero. At $a=1$, these derivatives give moments used in logarithmic endpoint <asymptotic expansions>.
Back to article page
