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We prove an asymptotic formula for the partition function of a 2d Coulomb gasat inverse temperature $\beta>0$ confined to lie on a Jordan curve. This alsogives a central limit theorem for a linear statistic of the particles in thegas. We obtain different expressions for the asymptotic mean and variance whichinvolve either the exterior conformal mapping of the curve or the Grunskyoperator.

Partition function for the 2d Coulomb gas on a Jordan curve