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Eddy-current probe impedance variations due to interactions with planar cracks have been calculated for the thin-skin regime. In this regime, the skin depth of the induced current is small compared to the crack depth and length, allowing approximations to be made. The approximations have been used by others to show that the thin-skin field at the surface of a crack is governed by a potential satisfying the two-dimensional (2-D) Laplace equation. In fact, the transverse magnetic potential at the crack face, defined with respect to the normal to this surface, satisfies a 2-D Laplace equation at an arbitrary skin depth. However, thin-skin boundary conditions applied at the crack perimeter greatly simplify the problem. Solutions of the Laplace problem for semielliptical cracks have been found by conformal mapping to a rectangular region. The surface potential in the rectangular domain is expressed as a Fourier series expansion and the coefficients of the series determined from the boundary conditions. Curved crack profiles of a general class, including semielliptic cracks as a special case, have been approximated by using ordered elliptical epicycles, a representation that retains the ability to map the crack domain to a rectangle. The probe impedance change due to a crack has been expressed in terms of the transverse magnetic potential and calculated from a line integral. Predictions of the probe impedance variations with position and frequency have been compared with an analytical solution for a semicircular crack and with experimental coil impedance measurements on semielliptical and epicyclic slots. Good agreement is observed in all comparisons.