the min. value of gamma function

**simplependulum** · August 10th 2009, 04:51 AM

What is the min. value of gamma function $\Gamma(x) ~~ x > 0$ ?

How do I start with .. finding zero derivative $\int_0^{\infty} e^{-t} \ln(t) t^{x-1}~ dt~ = 0$ ?

**Opalg** · August 10th 2009, 09:27 AM

Originally Posted by simplependulum

What is the min. value of gamma function $\Gamma(x) ~~ x > 0$ ?

How do I start with .. finding zero derivative $\int_0^{\infty} e^{-t} \ln(t) t^{x-1}~ dt~ = 0$ ?

The answer is given here (without explanantion) as $\Gamma(x_{\min}) = 0.885603194410888...$ . There doesn't seem to be a closed form expression for either $x_{\min}\ (= 1.461632144968362341262...)$ or $\Gamma(x_{\min})$ .

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