{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:13:23Z","timestamp":1776802403722,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"307","license":[{"start":{"date-parts":[[2017,9,27]],"date-time":"2017-09-27T00:00:00Z","timestamp":1506470400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100003329","name":"Ministerio de Econom\u00c3\u00ada y Competitividad","doi-asserted-by":"publisher","award":["MTM2014-52859"],"award-info":[{"award-number":["MTM2014-52859"]}],"id":[{"id":"10.13039\/501100003329","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We can find in the literature several convergent and\/or asymptotic expansions of the Pearcey integral\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P left-parenthesis x comma y right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">P(x,y)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in different regions of the complex variables\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"y\">\n                        <mml:semantics>\n                          <mml:mi>y<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">y<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , but they do not cover the whole complex\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"y\">\n                        <mml:semantics>\n                          <mml:mi>y<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">y<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    planes. The purpose of this paper is to complete this analysis giving new convergent and\/or asymptotic expansions that, together with the known ones, cover the evaluation of the Pearcey integral in a large region of the complex\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x\">\n                        <mml:semantics>\n                          <mml:mi>x<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">x<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"y\">\n                        <mml:semantics>\n                          <mml:mi>y<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">y<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    planes. The accuracy of the approximations derived in this paper is illustrated with some numerical experiments. Moreover, the expansions derived here are simpler compared with other known expansions, as they are derived from a simple manipulation of the integral definition of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper P left-parenthesis x comma y right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>P<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">P(x,y)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3164","type":"journal-article","created":{"date-parts":[[2016,4,14]],"date-time":"2016-04-14T12:47:03Z","timestamp":1460638023000},"page":"2399-2407","source":"Crossref","is-referenced-by-count":7,"title":["Analytic formulas for the evaluation of the Pearcey integral"],"prefix":"10.1090","volume":"86","author":[{"given":"Jos\u00e9","family":"L\u00f3pez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pedro","family":"Pagola","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,9,27]]},"reference":[{"key":"1","isbn-type":"print","first-page":"775","article-title":"Integrals with coalescing saddles","author":"Berry, M. 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A","ISSN":"https:\/\/id.crossref.org\/issn\/0962-8444","issn-type":"print"},{"issue":"1-2","key":"11","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1016\/j.cam.2005.01.038","article-title":"Hyperasymptotic evaluation of the Pearcey integral via Hadamard expansions","volume":"190","author":"Paris, R. B.","year":"2006","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"12","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1080\/14786444608561335","article-title":"The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic","volume":"37","author":"Pearcey, T.","year":"1946","journal-title":"Philos. Mag. (7)","ISSN":"https:\/\/id.crossref.org\/issn\/0031-8086","issn-type":"print"},{"key":"13","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1017\/s0305004100050945","article-title":"Integrals with a large parameter. Several nearly coincident saddle-points","volume":"72","author":"Ursell, F.","year":"1972","journal-title":"Proc. Cambridge Philos. 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