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.

15 pages, 2 figures. Extended discussion of some points