in this link "Sharp Smith’s bounds for the gamma function " I want to know with explain (Theorem 2) how that formula is got?
\begin{eqnarray*} g(x)=\ln \Gamma(x+\frac{x}{2})-[\frac{x}{2}\ln(2\pi)+\frac{16}{21}x\ln x+\frac{5x}{42}(x^2+\frac{7}{20})-x-\frac{\sqrt(35)}{42}\operatorname{arccot}(\frac{\sqrt(20)}{7}x)] \end{eqnarray*} How this function is built?