This paper presents the Maxwell model in simple shear flow by using the complex analysis, instead of the matrix analysis which is generally used in the literature. It is found that the viscoelastic fluids will behave viscoelastically only when the elastic shear deformation is significant, say about unity. Analysis of the viscoelastic flow based on the assumption of small elastic deformation will overlook the viscoelasticity. Because the elastic deformation is always great when the viscoelasticity is observed, the constant elasticity assumption, rather than the constant viscosity assumption, will lose its effect. This paper first introduces the assumption that the shear modulus of viscoelastic fluids is linearly related to the shear strain rate, and the equivalent viscosity is compared with the experimental results for some simple hydrocarbons in the literature. The theory proposed in this paper gives predictions agreed with the experimental results as well as the Carreau model.