  We derive an exact closed-form representation for the Euclidean thermal Green
function of the two-dimensional (2D) free massless scalar field in coordinate
space. This can be interpreted as the real part of a complex analytic function
of a variable that conformally maps the infinite strip $-\infty<x<\infty$
($0<\tau<\beta$) of the $z=x+i\tau$ ($\tau$: imaginary time) plane into the
upper-half-plane. Use of the Cauchy-Riemann conditions, then allows us to
identify the dual thermal Green function as the imaginary part of that
function.
