  The QCD baryon number density can formally be expanded into a Laurent series
in fugacity, which is a relativistic generalization of Mayer's cluster
expansion. We determine properties of the cluster expansion in a model with a
phase transition and a critical point at finite baryon density, in which the
Fourier coefficients of the expansion can be determined explicitly and to
arbitrary order. The asymptotic behavior of Fourier coefficients changes
qualitatively as one traverses the critical temperature and it is connected to
the branch points of a thermodynamic potential associated with the phase
transition. The results are discussed in the context of lattice QCD simulations
at imaginary chemical potential. We argue that the location of a branch point
closest to the imaginary chemical potential axis can be extracted through an
analysis of an exponential suppression of Fourier coefficients. This is
illustrated using the four leading coefficients both in a toy model as well as
by using recent lattice QCD data.
