Gamma function properties

Question

Hello everybody can I ask about this property :

For $z$ small between 0 and 1, $ {\mathrm{(}}\mathit{\Gamma}{\mathrm{(}}{z}{\mathrm{))}}^{\mathrm{{-}}{1}}\mathrm{\approx}{z}. $

Is there any explanation . Thanks a lot in advance.

Answer 1

$$z\Gamma(z)=\Gamma(z+1)$$ $$z\to 0\quad\implies\quad \Gamma(z+1)\approx \Gamma(1)=1$$ $$ z\Gamma(z)\approx 1$$ $$\frac{1}{\Gamma(z)}\approx z$$ For more accurate approximates, see : https://en.wikipedia.org/wiki/Gamma_function#General

Answer 2

$\Gamma'(1)=-\gamma$, so for small $x$, $\Gamma(1+x)\approx1-\gamma x$. Thus, $$ \Gamma(x)=\tfrac{\Gamma(1+x)}x\approx\frac1x-\gamma $$ where $\gamma=0.5772156649$ is the Euler-Mascheroni constant. Therefore, $$ \frac1{\Gamma(x)}\approx x+\gamma x^2 $$