Thermal Green's functions of the fluctuation electromagnetic field of a cylinder and a cylindrical layer embedded in an infinite medium

The exact thermal Green's functions for the long-wavelength fluctuation electromagnetic field are found for a cylinder of radius $R_1$ with permittivity $\varepsilon_1$ surrounded by a layer of radius $R_2$ with permittivity $\varepsilon_2$ embedded in an infinite medium with permittivity $\varepsilon_3$. It is shown how these Green's functions can be used to find various thermodynamic characteristics of such a system. As examples, the chemical potential of a cylindrical film and the pressure produced by it and also the change in the density of a liquid near a cylindrical interface are considered. In the general case, the obtained expressions can be used for computer calculations. In important limiting cases, simple analytic dependences are obtained.