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The analytical approximate technique developed by Wu et al for conservative oscillators with odd non-linearity is used to construct approximate frequency-amplitude relations and periodic solutions to the relativistic oscillator. By combining Newton's method with the method of harmonic balance, analytical approximations to the oscillation period and periodic solutions are constructed for this oscillator. The approximate periods obtained are valid for the complete range of oscillation amplitudes,doi:10.1142/s0217979209049954 fatcat:u2jbcuxe6fdrxmoe3rxdoss4mu