We give an explicit formula for the fundamental solutions of an elliptic linear partial differential operator of the second order with analytic coefficients and simple complex characteristics in an open set &Omega; &sub; R<sup>n</sup>. We prove that those fundamental solutions can be continued at least locally as multi-valued analytic functions x &rarr; E(x, y) in C<sup>n</sup> up to the complex bicharacteristic conoid. This extension ramifies along its singular set the bicharacteristic conoid and belongs to the Nilsson class.