  The self-duality transformation is applied to the Fisher zeroes near the
critical point in the thermodynamic limit in the q>4 state Potts model in two
dimensions. A requirement that the locus of the duals of the zeroes be
identical to the dual of the locus of zeroes (i) recovers the ratio of specific
heat to internal energy discontinuity at criticality and the relationships
between the discontinuities of higher cumulants and (ii) identifies duality
with complex conjugation. Conjecturing that all zeroes governing ferromagnetic
critical behaviour satisfy the latter requirement, the full locus of Fisher
zeroes is shown to be a circle. This locus, together with the density of zeroes
is shown to be sufficient to recover the singular form of all thermodynamic
functions in the thermodynamic limit.
