NOVELTY - Manufacture of lithium-containing complex oxide (I) involves obtaining a hydroxide containing nickel and manganese, mixing a lithium compound with the hydroxide, obtaining a mixture, heat-processing the mixture having oxygen concentration of 5 vol.% or more at 500-700 degrees C, obtaining a primary heat-treated substance, heat-processing the primary heat-treated substance at a temperature of 800-950 degrees C in an atmosphere having oxygen concentration of less than 5 vol.%, and obtaining lithium-containing complex oxide.    USE - Manufacture of lithium-containing complex oxide used for forming positive electrode of lithium-ion secondary battery (all claimed) for portable electronic device and as vehicle-mounted lithium-ion secondary battery.    ADVANTAGE - The method enables manufacture of lithium-containing complex oxide, which provides lithium-ion secondary battery with excellent cycle characteristics and high discharge capacity.    DETAILED DESCRIPTION - Manufacture of lithium-containing complex oxide of formula: Li1+aMOb (I) involves obtaining a hydroxide containing nickel and manganese, mixing a lithium compound with the hydroxide such that the molar ratio of lithium with respect to total number of moles of metallic elements other than lithium contained in the hydroxide reaches 1-1.6, obtaining a mixture, heat-processing the mixture having oxygen concentration of 5 vol.% or more at a temperature of 500-700 degrees C, obtaining a primary heat-treated substance, heat-processing the primary heat-treated substance at a temperature of 800-950 degrees C in an atmosphere having oxygen concentration of less than 5 vol.%, and obtaining lithium-containing complex oxide. The molar ratio of manganese with respect to nickel contained in the hydroxide is 1.1 or more. The hydroxide has a specific surface area of 5-80 m2/g.    M=metallic element chosen from nickel or manganese (excluding lithium);    a=0-0.6;and    b=number of moles of oxygen required for satisfying valence of lithium and M.