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The Laplace's equations for the scalar and vector potentials describingelectric or magnetic fields in cylindrical coordinates with translationalinvariance along azimuthal coordinate are considered. The series of specialfunctions which, when expanded in power series in radial and verticalcoordinates, in lowest order replicate the harmonic homogeneous polynomials oftwo variables are found. These functions are based on radial harmonics found byEdwin~M.~McMillan in his more-than-40-years "forgotten" article, which will bediscussed. In addition to McMillan's harmonics, second family of adjoint radialharmonics is introduced, in order to provide symmetric description betweenelectric and magnetic fields and to describe fields and potentials in terms ofsame special functions. Formulas to relate any transverse fields specified bythe coefficients in the power series expansion in radial or vertical planes incylindrical coordinates with the set of new functions are provided.  This result is no doubt important for potential theory while also criticalfor theoretical studies, design and proper modeling of sector dipoles, combinedfunction dipoles and any general sector element for accelerator physics. Allresults are presented in connection with these problems.

On sector magnets or transverse electromagnetic fields in cylindrical coordinates