# Minimum of the Gamma Function $\Gamma (x)$ for $x>0$. How to find $x_{\min}$?

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.

• A general rule of thumb in numerical computing: it's easier to compute (simple) roots of functions to the full precision of your environment than to compute extrema. – J. M. is a poor mathematician Aug 24 '10 at 22:18
• wolframalpha.com/input/?i=min+Gamma%28x%29+from+0+to+3 – Memming May 9 '12 at 15:33
• computation results in 1935 nature.com/nature/journal/v135/n3422/abs/135917b0.html – Memming May 9 '12 at 15:46
• @Memming Thanks for the link. – Américo Tavares May 9 '12 at 15:48
• So as it's not simple to compute the value analytically, it's worth stating that $\left(\frac{33}{20}\cdot\frac{\pi^4}{110}\ ,\ \frac{\pi^4}{110}\right)$ is an approximation to $(x,y)_{\text{min}}$ with an error of $\mathcal{O}(10^{-4})$ in the first and even less in the second variable. – Nikolaj-K Mar 1 '13 at 11:04

• For instance, the computation in Mathematica goes something like x /. FindRoot[PolyGamma[x], {x, 1}, WorkingPrecision -> 20] which yields the result 1.4616321449683623413 . – J. M. is a poor mathematician Aug 24 '10 at 22:11
According to MathWorld the minimum of the Gamma function for positive $x$ is 1.46163...; in particular I guess this is enough to deduce that it is smaller than $3/2$. You can follow the links along to find some references where this is proved.