So today I'll be holding a seminar on Euler's gamma function, as part of my uni work... I'm happy with my presentation but I am unsure what to answer if the question pops up "why isn't the gamma function defined for negative integers?"
One of the defining properties of $\Gamma$ is that $$ x\Gamma(x)=\Gamma(x+1) $$ Assuming $\Gamma(1)$ is strictly positive, what could $\Gamma(0)$ possibly be to respect this relation? We would want $0\Gamma(0)=\Gamma(1)$. That can't happen. So it must remain undefined there.
Now we can go backwards: If $\Gamma(-1)$ were defined, we would simply set $\Gamma(0)=-\Gamma(-1)$. But we know $\Gamma(0)$ can't have a value. So neither can $\Gamma(-1)$. And so on.
This is a bit handwavey, and not a real proof. But it should, at the very least, serve as evidence and justification.