Chapter A, Problem 27E

1. (a) For x > 0 the gamma function possesses derivatives of all orders. Use Leibniz’s rule in the form given in (24) to show that

   $\begin{array}{c}{\Gamma }^{\prime }(x)={\displaystyle {\int }_0^{\infty }{t}^{x-1}{e}^{-t}}\ln tdt\\ {\Gamma }^{″}(x)={\displaystyle {\int }_0^{\infty }{t}^{x-1}{e}^{-t}}{(\ln t)}^{2}dt.\end{array}$

2. (b) For x > 0, Γ(x) > 0 and Γ″(x) > 0. Without looking back at Figure A.1, what does this say about the graph of Γ(x)?