Exact high temperature expansion of the one-loop thermodynamic potential with complex chemical potential

We present a derivation of an exact high temperature expansion for a one-loop thermodynamic potential $\Omega(\tilde{\mu})$ with complex chemical potential $\tilde{\mu}$. The result is given in terms of a single sum the coefficients of which are analytical functions of $\tilde{\mu}$ consisting of polynomials and polygamma functions, decoupled from the physical expansion parameter $\beta m$. The analytic structure of the coefficients permits us to explicitly calculate the thermodynamic potential for the imaginary chemical potential and analytically continue the domain to the complex $\tilde{\mu}$ plane. Furthermore, our representation of $\Omega(\tilde{\mu})$ is particularly well suited for the Landau--Ginzburg-type of phase transition analysis. This fact, along with the possibility of interpreting the imaginary chemical potential as an effective generalized-statistics phase, allows us to investigate the singular origin of the $m^3$ term appearing only in the bosonic thermodynamic potential.