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On-surface radiation conditions are useful for obtaining approximate solutions to scattering problems involving compact obstacles. An analytic representation of the Dirichlet-to-Neumann map for a circle is derived and used to construct a higher-order on-surface radiation condition for a generally convex perfectly conducting body in two dimensions. This approach is based on a Hankel function in which a tangential operator appears in the index. In the high-frequency limit, this analytic representation approaches the square root of a differential operator which commonly arises in the application of parabolic equation techniques to propagation problems. Treating the scattered field propagation angle relative to the surface normal and the surface curvature as independent parameters, the representation is fit to a rational function to provide an accurate and efficient on-surface radiation condition that is tested for various examples.