  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.
