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A theoretical analysis of an antenna consisting of a perfectly conducting semi-infinite cone excited by a thin, linear element directed along its axis is presented. Two different methods are applied; one is a Green's function method, and the other is a mode theory. By the former methods the radiation characteristics of very thin and wide-angle cones excited by a sinusoidal current element are obtained. In the latter methods the linear element at the tip is replaced by a finite cone resulting in a semi-infinite, biconical structure. The variational principle is applied to determine the effective terminating admittance, and thus the input impedances and the far-zone field. A guided wave which propagates longitudinally on the semi-infinite cone is identified and its properties are discussed.