# Math Help - The Gamma Function

1. ## The Gamma Function

Hello,

I'm trying to teach myself the Gamma Function from here:

The Gamma Function

However, I'm a tad confused at some parts.

I don't understand at all how they figured out (derived) the inequality where it says

"A careful analysis of the Gamma function (especially if we notice that is a convex function) yields the inequality"

...and I don't know what a convex function is. I don't really see the significance in deriving the function in all these different ways.

Also, when they give the intial formula, I don't understand how it was derived/where it came from. Does anyone know how it was formulated?

$\int^{\infty}_0 e^{-t}t^n\,dt=n!$

3. If you represent the $\Gamma(*)$ as ‘infinite product’…

$\Gamma (x) = \frac{e^{-\gamma x}}{x} \cdot \prod_{n=1}^{\infty} \frac{e^{\frac{x}{n}}}{1+\frac{x}{n}}$ (1)

… then the $\ln \Gamma(*)$ is simply the sum of the logarithms of all factors…

$\ln \Gamma (x) = -\gamma x - \ln x + \sum_{n=1}^{\infty} \{\frac{x}{n} - \ln (1 + \frac{x}{n})\}$ (2)

Kind regards

$\chi$ $\sigma$