  Although partition functions of finite-size systems are always analytic, and
hence have no poles, they can be expressed in many cases as series containing
terms with poles. Here we show that such poles can be related to linear
branches of the entropy, expressed in the thermodynamic limit as a function of
the energy per particle. We also show that these poles can be used to determine
whether the entropy is nonconcave or has linear parts, which is something that
cannot be done with the sole knowledge of the thermodynamic free energy derived
from the partition function. We discuss applications for equilibrium systems
having first-order phase transitions.
