Phase transistions and Partition functions

I'm studying phase transitions in statistical mechanics, and I'm curious to know more about how the partition function becomes zero. I've read a bit about the Lee Yang theorem, but haven't found a good description of the whole program. Does anyone have a good paper on the subject? (Other than Lee and Yang's?)
Also, I can't figure out if the partition function itself is analytic in the sense that it can be expanded about some neighborhood and is infinitely differentiable. Can anyone shed some led on the analytic properties of partition functions?
Thanks