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This paper presents the contour integral method for solving the linear constant coefficient ordinary differential equa-tions in complex plane,and obtains the uniform expressions of the general solutions.Firstly,by using Residue Theorem,the general form of the contour integral representation for the homogeneous complex differential equation is obtained,which can be degener-ated to classical results in real line.As for inhomogeneous com-plex differential equations with constant coefficients,we construct the integral expression of the particular solution for any continu-ous forcing term,and give rigorous proof via Residue Theorem.Thus the general solutions of inhomogeneous complex differential equations are also given.The main purpose of this paper is to give a foundation for a complete theory of linear complex differential equations with constant coefficients by a contour integral method.The results can not only solve the inhomogeneous complex dif-ferential equation well,but also explain the forms that are difficult to be understood in the classical solutions.