It is widely known that $0^0$ is usually defined to be 1. I wonder why we cannot employ a similar technique to ascribe values to functions having poles in a point.
Now take the Gamma function. The real part $\Re(\Gamma(i x))$ is equal to $\gamma$ (Euler-Masceroni constant). We thus can use the formulas:
$$\Gamma(-n)=\lim_{h\to0} \Re (\Gamma(-n+ih))$$
or, alternatively,
$$\Gamma(-n)=\lim_{h\to0} \frac{\Gamma(-n+h)+\Gamma(-n-h)}2$$
Thus we can ascribe natural values to Gamma function at negative integers: $$\Gamma(0)=-\gamma$$ $$\Gamma(-1)=\gamma-1$$ $$\Gamma(-2)=\frac{3}{4}-\frac{\gamma }{2}$$ $$\Gamma(-3)=\frac{\gamma }{6}-\frac{11}{36}$$ $$\Gamma(-4)=\frac{25}{288}-\frac{\gamma }{24}$$
etc.
I wonder why these values are not given in tables and not used in computer algebra systems, this could symplify things a lot.