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The time dependent Green’s function boundary problem for a semi-infinite perfectly conducting slotted cone placed on the impedance plane is considered. This geometry can be regarded as a suitable three-dimensional model of slotted cone antennas and wire antennas situated on the imgomogenous plane. The solution method is based on using the Laplace transform, the Kontorovich-Lebedev transform and the Rieman-Hilbert method. The Green’s function representations are derived for the “partly penetrable” cone. The boundary problem spectrum and the plane contribution are investigated.