Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the Pearcey integral P(x, y) for large values of |y| and bounded values of |x|. The integrand of the Pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is complicated. Then  consider here the modified saddle point method introduced in Lopez, Perez and Pagola, 2009 4. With this method, the analysis is simpler and it is possible to derive a complete asymptotic expansion of P(x, y) for large |y|. The asymptotic analysis requires the study of three different regions for arg y separately. In the three regions, the expansion is given in terms of inverse pors of y2/3 and the coefficients are elementary functions of x. The accuracy of the approximation is illustrated with some numerical experiments.
['José L. López', 'Pedro J. Pagola']
I am writing a research paper on a idea The Pearcey integral in the highly oscillatory region
[{"from": "human", "value": "I am writing a research paper on a idea The Pearcey integral in the highly oscillatory region"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  consider the Pearcey integral P(x, y) for large values of |y| and bounded values of |x|. The integrand of the Pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is complicated. Then  consider here the modified saddle point method introduced in Lopez, Perez and Pagola, 2009 4. With this method, the analysis is simpler and it is possible to derive a complete asymptotic expansion of P(x, y) for large |y|. The asymptotic analysis requires the study of three different regions for arg y separately. In the three regions, the expansion is given in terms of inverse pors of y2/3 and the coefficients are elementary functions of x. The accuracy of the approximation is illustrated with some numerical experiments."}]