Abstract: In this paper, the general mathematical principle is over-all explained and a new general technique is presented in order to calculate uniformly asymptotic expansions of solutions of the perturbed bifurcation problem (1.6) in the vicinity of y=0, λ=0,δ=0, by means of singular perturbation method. Simultaneously, Newton’s polygon^[4] is generalized. Finally, the calculating results of two examples are given.

Key words: theta(t)-type oscillatory singular integral, weighted norm inequality, Hardy-Littlewood maximal operator