Behaviour of the Gamma function near zero

Question

The Gamma function on the positive real half-line is defined via the reknown formula $$ \Gamma(z)=\int_0^\infty x^{z-1}e^{-x}dx, \quad z>0. $$ A classical result is Stirling's formula, describing the behaviour of $\Gamma(z)$ as $z$ diverges to infinity, $$ \Gamma(z)\sim \sqrt{\frac{2\pi}{z}} \left( \frac{z}{e}\right)^z, \quad z \to \infty. $$ Is there any such approximation formula for $z \downarrow 0$, describing the speed at which the Gamma function diverges near the origin?

Answer 1

Around $z=0$ $$\Gamma(z)=\frac{1}{z}-\gamma +\sum_{n=1} a_n z^n$$ $$a_1=\frac{1}{12} \left(6 \gamma ^2+\pi ^2\right)\qquad a_2=-\frac{1}{6} \left(2 \zeta (3)+\gamma ^3+\frac{\gamma \pi ^2}{2}\right)$$ $$a_3=\frac{1}{24} \left(8 \gamma \zeta (3)+\gamma ^4+\gamma ^2 \pi ^2+\frac{3 \pi ^4}{20}\right)$$

Edit

A quite good approximation is given by $$\Gamma(z)\sim \frac 1 z \frac{1+\frac{\left(\pi ^2-6 \gamma ^2\right) }{12 \gamma }z } {1+\frac{\left(\pi ^2+6 \gamma ^2\right) }{12 \gamma }z }$$