Analytic solutions to the Laplace equation in annulus and disc are combined with transmission conditions to find analytic solutions for transmission problems with multiple concentric circular interfaces. Coefficients are found rapidly by solving 2 × 2 linear systems for each interface. This makes them very suitable as test cases for inverse problem solvers. The special case of two very close interfaces is used to quantitatively test crack jump conditions which result from combining transmission conditions at the two interfaces into approximate conditions at a single interface. The quality of the approximation depends on whether the crack is resistive or conductive.