Rigorous proof for analyticity of the homomorphic cluster coherent potential approximation

We rigorously prove that the homomorphic coherent potential approximation (HCPA) is analytic. Along a way analogous to the proof for the analyticity of the molecular coherent potential approximation by Ducastelle, we show the HCPA always provides a physical solution for the average Green's function which satisfies (i) the reality, (ii) definiteness, (iii) analyticity and (iv) boundary conditions.