摘要：This paper concerns about the equivalence between the Melnikov method and the averaging method for studying the number of limit cycles, which are bifurcated from the period annulus of planar analytic differential systems. We prove that for any positive integer k, the kth Melnikov function is equal to the kth averaging function with a 2_ multiplication modulo both Melnikov and averaging functions of order less than k, which appear in planar analytic near–Hamiltonian systems.