  The 2d one component gas of pointlike charges in a uniform neutralizing
background interacting with a logarithmic potential is a common model for
plasmas. In its classical equilibrium statistics at fixed temperature
(canonical ensemble) it is formally related to certain types of random matrices
with Gaussian distribution and complex eigenvalues. In this paper, I present an
exact integration of this ensemble for $N$ such particles (or alternatively
$N\times N$ matrices) for all even non-negative temperatures, a significant
open problem in statistical physics for several decades. I achieve this exact
integration via an exact integration of a related ensemble, the two-dimensional
Selberg integral.
