> Sure enough, we see that the gamma function has no limit at > the nonpositive integers, always approaching positive > infinity from one side and negative in

> Sure enough, we see that the gamma function has no limit at > the nonpositive integers, always approaching positive > infinity from one side and negative infinity from the other side. On the complex plane, there is more than one side to approach from. And the convention that was used when I learned complex analysis (stereographic projection), is that there is only one value of infinity, which can be approached from many directions. Under this model, the point at infinity is much better-behaved than the points at infinity in real analysis. The Gamma function is characterized as having poles at the negative integers. That's a very straightforward description, and the behavior is well-understood and relatively easy to deal with. Certainly easier to deal with than the value of e^-x as x->0 (that singularity is essential, rather than a pole).