  The momentum plane analyticity of two point function of a relativistic
thermal field theory at zero chemical potential is explored. A general
principle regarding the location of the singularities is extracted. In the case
of the N=4 supersymmetric Yang-Mills theory at large $N_c$, a qualitative
change in the nature of the singularity (branch points versus simple poles)
from the weak coupling regime to the strong coupling regime is observed with
the aid of the AdS/CFT correspondence.
