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The formulas required for simple accurate numerical evaluation of the radiation patterns from slots on large circular cylinders are derived. From the usual harmonic series, integral representations are obtained. These lead to either the residue series via the Watson transformations or to the geometrical-optics representations via a saddle-point evaluation. Considerable attention is given to the calculation of the residue series constants so as to obtain sufficiently accurate numerical values for a wide range of circumference to wavelength ratios. Very good numerical agreement is shown between the representations derived here and results previously obtained from the harmonic series. The physical significance of the various representations is discussed and compared.