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In the course of designing and analyzing electromagnetic devices, open boundary field problems often need to be solved. The mapping of the infinite exterior of a finite circular region onto the interior of that closed circular region has emerged and established itself as one of the best methods. However, as a result of the work done in the past in solving homogenous exterior regions, this paper shows that this mapping method has been mistakenly taken to apply only when the exterior consists of air or, at best, some other homogeneous medium. This paper validates the method for problems with uniformly inhomogeneous exterior regions and makes possible the solution of a new class of problems by this powerful method.