Fredholm series solution to the integral equations of thermal blooming

From the solution for the linear theory of thermal blooming, the propagator is a 2 X 2 matrix that satisfies an integral equation of Fredholm type. We develop a generalized Fredholm series solution to this integral equation. Since the Kernel is a matrix, the usual determinants in the Fredholm series contain ordering ambiguities. We resolve all ordering ambiguities using the standard diagrammatic representation of the series. The Fredholm denominator is computed for the case of uncompensated and compensated propagation in a uniform atmosphere with uniform wind. When the Fredholm denominator vanishes, the propagator contains poles. In the compensated case, the denominator does develop zeros. The single mode phase compensation instability gains computed from the zeros agrees with results obtained from other methods.