The conformal mapping w=(L/2π)\ln z transforms the critical plane with a radial perturbation αρ^{-y} into a cylinder with width L and a constant deviation α(2π/L)^y from the bulk critical point when the decay exponent y is such that the perturbation is marginal. From the known behavior of the homogeneous off-critical system on the cylinder, one may deduce the correlation functions and defect exponents on the perturbed plane. The results are supported by an exact solution for the Gaussian model. ; Comment: Old paper, for archiving. 3 pages, RevTeX