= Two-region domination for differentiation of the gamma integral
{title2=$|\partial_t(e^{-x}x^{t-1})|\leq x^{a-1}|\log x|\mathbf1_{x\leq1}+e^{-x}x^{b-1}\log x\mathbf1_{x>1}$}
For $t$ in a compact positive interval $[a,b]$, split the integration region at one. Near zero the smallest exponent gives the displayed bound, whose integral is $1/a^2$. Above one the largest exponent gives the bound, integrable by <exponential decay>. A single <integrable function> therefore dominates all parameter derivatives locally. <Dominated differentiation under the integral sign> gives $\Gamma'(t)=\int_0^\infty e^{-x}x^{t-1}\log x\,dx$. The same argument with higher powers of $|\log x|$ gives all higher derivatives.
Back to article page
