Mathematical properties of -curve in the frame work of the homotopy analysis method

As it is described in the frame work of the homotopy analysis method (HAM), the convergence-control parameter is the main auxiliary tool which distinguishes this method form the other analytical methods. Moreover the convergence is usually obtained by the so-called ℏ -curve which possesses horizontal line property. The purpose of this paper is to answer this fundamental question: That is, why the horizontal line occurs in the plot of HAM series solution at some points corresponding to the convergence-control parameter. Also, the mathematical proof and the properties of this main issue are presented. Furthermore, some illustrative examples are presented and the salient features are discussed.