Differential quotient rule of composite function operator and its applications in quantum physics, quantum statistics, operator ordering theory, matrix theory and control theory are given. The integration problem of Wigner operator and Weyl corresponding rules are studied. Two kinds of typical operator identity formulas are proved. The differential form of Wigner operator in ordered product of operators and new differential form of important functions are obtained. Finally, a Wigner operator with parameter for unifying regular order, Weyl sequencing and abnormal order is introduced.