Theory

. This paper gives an introduction to the theory of oscillatory integrals of the first kind. Due to the complexity and multiplicity of critical points in higher dimensions, the decay of | I ( λ ) | in R d with d > 1 is far from clear-cut. However, using the method of stationary phase, one can establish an explicit decay property of the oscillatory integral in certain cases. We will first prove decay of | I ( λ ) | in the more straightforward case in which the phase function does not have a critical point. Next, we will analyze the case of a nondegenerate critical point and demonstrate some applications of oscillatory integrals in the study of partial differential equations.