In this paper, a multilayered cylinder filled with double negative (DNG) material and double positive (DPS) material is studied. General formulas of electromagnetic fields in each region are derived using the eigenfunction expansion method. The expansion coefficients are determined by a recursive system that is derived from boundary conditions at the multiple interfaces. The reflection by a cylinder with (-$varepsilon$ 0, -5 0) (where $varepsilon$ 0 and 5 0 denote the permittivity and permeability in free space, respectively) is found to be quickly diminished with its electric size. For an infinite line source, the image property is observed when the radius of this cylinder is much larger than the wavelength. The distributions of electromagnetic fields are presented when a line source is placed near a two-layered cylinder alternately filled with DNG and DPS material. Numerical results confirm that the developed formulas are suitable to analyze the other multilayered cylindrical structures with DNG and DPS materials.