  We consider the conductivity problem in the presence of adjacent circular
inclusions having arbitrary constant conductivity. When two inclusions get
closer and their conductivities degenerate to zero or infinity, the gradient of
the solution can be arbitrary large. We characterize the gradient blow-up by
deriving an explicit formula for the singular term of the solution in terms of
the Lerch transcendent function. This derivation is valid for inclusions having
arbitrary constant conductivity. We illustrate our results with numerical
calculations.
