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We derive an exact closed-form representation for the Euclidean thermal Greenfunction of the two-dimensional (2D) free massless scalar field in coordinatespace. This can be interpreted as the real part of a complex analytic functionof 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 theupper-half-plane. Use of the Cauchy-Riemann conditions, then allows us toidentify the dual thermal Green function as the imaginary part of thatfunction.

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