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Analytical solutions for smooth positive-to-negative transition materials

We obtain analytical solutions for positive-to-negative transition materials with a smooth transition profile "$\tanh$". The fields are expressed in terms of hypergeometric functions. The final expressions for reflection and transmission coefficients are obtained in closed forms. The total absorption is nonzero even in the lossless limit in accordance with previous studies. Properties of the total absorption as function of controlling factors are also well studied. As an application of the analytical results, we analyze the effects of smooth transition on perfect imaging and, interestingly, find multiple images of the object.