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A VERTICAL electric dipole in the plane interface between two semi-infinite homogeneous dielectric media is switched on at . The resulting electromagnetic field is determined by a new method based on a new representation of the field of a point source as the superposition of the fields of line sources. The field of a line dipole is found by using, first, the conical flow transformation, and then a Jonkowski transformation which leads to a function-theoretic problem of the Riemann-Hilbert type. This problem is solved explicitly and an explicit expression is obtained for the point dipole field at points of the interface. The results are finally compared with those of others.