The $\Gamma (x)$ function has just one minimum for $x>0$ . This result uses some properties of the gamma function:
- $\Gamma ^{\prime \prime }(x)>0$ and $\Gamma (x)>0$ for all $x>0$
- $\Gamma (1)=\Gamma (2)=1$.
Observing the following graph (created in SWP) of $y=\Gamma (x)$ this minimum is near $x=3/2$, but likely $\min \Gamma (x)\neq \Gamma \left( 3/2\right) =\dfrac{1}{2}\Gamma \left( 1/2\right) =\dfrac{1}{2}\sqrt{\pi }$.
I think that it is not possible to find analytically the exact value of $x_{\min }$, even by converting to an adequate problem in the interval $]0,1]$ and using the functional equation $\Gamma (x+1)=x\Gamma (x)$ and the reflection formula
$\Gamma (p)\Gamma (p-1)=\dfrac{\pi }{\sin px}\qquad $( $0\lt p\lt 1$)
Question:
a) Which is the best way to find $\min_{[1,2]}\Gamma (x)$ and does $x_{\min }$ lay in $[1,3/2]$ or in $[3/2,2]$?
b) Is there some useful series expansion of $\Gamma (x)$?
c) Which numeric method do you suggest?
Edit: Due to the shape of $\Gamma (x)$ I thought on the one-dimensional Davies-Swann-Campey method of direct search for unconstrained optimization, which approximates a function near a minimum by successive approximating quadratic polynomials.