  We give explicit integral formulas for the solutions of planar conjugate
conductivity equations in a circular domain of the right half-plane with
conductivity $\sigma(x,y)=x^p$, $p\in\mathbb{Z}$. The representations are
obtained via a Riemann-Hilbert problem on the complex plane when $p$ is even
and on a two-sheeted Riemann surface when $p$ is odd. They involve the
Dirichlet and Neumann data on the boundary of the domain. We also show how to
make the conversion from one type of conditions to the other by using the
so-called global relation. The method used to derive our integral
representations could be applied in any bounded simply-connected domain of the
right half-plane with a smooth boundary.
