  Some consequences of the analyticity of the free energy (pressure) of QCD at
finite chemical potential are deduced. These include a method for numerical
exploration of the full phase diagram by a novel use of simulations at
imaginary chemical potential to extract Yang-Lee zeroes of the grand-canonical
partition function. We make use of, and comment on, CPT symmetries, positivity
of non-linear susceptibilities and the finiteness of screening lengths. We also
comment on the structure of zeroes expected for the usual picture of the phases
of QCD, following a discussion of the physics of imaginary chemical potential.
