Search for Combinations of Thermal n-point Functions with Analytic Extensions

The 2^{n} different n-point functions that occur in real-time thermal field theory are Fourier transformed to real energies. Because of branch cuts in various energy variables, none of these functions can be extended analytically to complex energies. The known linear combinations that form the fully retarded and advanced functions can be extended analytically. It is proven that no other linear combinations have an analytic extension to complex energies.