The convergence of' Green's Junction expansions used in the exact analytical treatment @"problems involving boundaries qfd@rent shapes is a property crucial in obtaining their solution. Existing expansions in most cases sufler front two serious setbacks : they do not conver~ge uniformly in their region oj'validity, exhibiting a slow and conditional convergence near the source (singular) point and, even worse, they change expression w>hen thejield point moves past the source point. For such reasons they are unsuited,for the solution of singular integral equations, in which values of the Grc~pn'.s,func,tion G at the source point do appear inside the integral. These inadequacies are met head-on by extracting the singular behavior in a closed:form term. Additional simple terms are also extracted to improve the convergence of the expansion of the remaining, non-singular part of' G. The so-obtained new ei~qenjiinction expansions,for G converge uniformly over the whole region oj their validity and very strongly (exponentially) near the source point. They are particularly suited.for the solution of singular integral equations by the Carleman-Vekua method, otherwise known as the method of regularization by solving the dominant equation. These new expansions can befurther subjected to a Watson transjormation yielding a third expansion exhibiting strong convergence in regions where the convergence of the preceding series weakens, and vice versa. All these