Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too!

The original motivation for this is the (finite) product $f(n)=\prod_{i=1}^{n-1}(1-\frac{i}{i^2+n})$ that I had to bound some time ago. Applying some calculus (logarithm to convert into sum, relate to a series, bound with an integral) I could show that $f(n)>\frac{1}{\sqrt{n}}-\frac\pi{8n}$ if $n>9$ (and actually $f(n)\sim \frac{1}{\sqrt{n}}-\frac\pi{8n}$ for big $n$) but I was left with the question whether there is a closed form for "my" finite product, or for the corresponding infinite product $\phi(n)=\prod_{i\ge 1}(1-\frac{i}{i^2+n})$(Any information on it would also make my day).

**EDIT May 17**: the infinite product $\phi(n)$ is zero (see Robert Israel's answer). Nevertheless, the square of $f(n)$ is $(1/n)\prod_{i=1}^{n-1}(1-(\frac{i}{i^2+n})^2)$, and it is still possible that the infinite product $\prod_{i\ge1}(1-(\frac{i}{i^2+n})^2)$ converges.

So, is there a place to look for techniques to deal with such products if the need arises?

Infinite Series, has a lot about products as well as sums [but disclaimer: I haven't even looked at this book for at least 15 years!] You might have some luck with various English mathematics books written before 1950, since this kind of thing was a lot more in fashion back then, at least in England [presumably due to G.H.Hardy's influence, although I am no historian]. – Zen Harper May 17 '11 at 6:23