The Pearcey integral in the highly oscillatory region

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• 作者：José ; L. Ló ; pez ; ^a ; ^{jl.lopez@unavarra.es" class="auth_mail" title="E-mail the corresponding author} ; Pedro J. Pagola^b ; ^{pedro.pagola@unavarra.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Pearcey integral ; Asymptotic expansions ; Modified saddle point method
• 刊名：Applied Mathematics and Computation
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：275
• 期：Complete
• 页码：404-410
• 全文大小：435 K

文摘

We consider the Pearcey integral m>Pm>(m>x, ym>) for large values of |m>ym>| and bounded values of |m>xm>|. The integrand of the Pearcey integral oscillates wildly in this region and the asymptotic saddle point analysis is complicated. Then we consider here the modified saddle point method introduced in [Lopez, Pérez and Pagola, 2009] [4]. With this method, the analysis is simpler and it is possible to derive a complete asymptotic expansion of m>Pm>(m>x, ym>) for large |m>y  m>|. The asymptotic analysis requires the study of three different regions for mmlsi14" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0096300315015829&_mathId=si14.gif&_user=111111111&_pii=S0096300315015829&_rdoc=1&_issn=00963003&md5=305f849e1a1d7f113c4627ac2a6d8bf9" title="Click to view the MathML source">argymathContainer hidden">mathCode"><math altimg="si14.gif" overflow="scroll"><mrow><mi>argmi><mi>ymi>mrow>math> separately. In the three regions, the expansion is given in terms of inverse powers of m>ym>^2/3 and the coefficients are elementary functions of m>xm>. The accuracy of the approximation is illustrated with some numerical experiments.