Chapter A, Problem 27E

(a)

To determine

To show: The equation ${\Gamma }^{\prime }\left(x\right)={\displaystyle {\int }_0^{\infty }{t}^{x-1}{e}^{-t}\ln tdt}$ and ${\Gamma }^{″}\left(x\right)={\displaystyle {\int }_0^{\infty }{t}^{x-1}{e}^{-t}{\left(\ln t\right)}^{2}dt}$ by the use of gamma function and Leibniz’s rule for $x>0$.

(b)

To determine

The behavior of the graph of a $\Gamma \left(x\right)$, if for $x>0$, $\Gamma \left(x\right)>0$ and ${\Gamma }^{″}\left(x\right)>0$.