Obtaining a closed-form representation for the dual bosonic thermal Green function by using methods of integration on the complex plane

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 $-\infty<x<\infty$ ($0<τ<β$) of the $z=x+iτ$ ($τ$: 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.

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