Expressions are derived for the external field produced by a slot of arbitrary shape in the wall of a circular wave guide (of infinite extent and infinite conductivity), the tangential components of the electric field in the slot being assumed to have been prescribed. This is accomplished by matching a Fourier representation of the external field, built up by superposition of basic sets of cylindrical waves, to a Fourier expansion of the prescribed field over the surface of the cylindrical wave guide. The far‐zone field is obtained by applying the method of steepest descent to the Fourier integrals that constitute the coefficients in the expansion for the external field. The results satisfy the radiation conditions for far‐zone fields.