{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:16:59Z","timestamp":1776802619825,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"309","license":[{"start":{"date-parts":[[2018,5,31]],"date-time":"2018-05-31T00:00:00Z","timestamp":1527724800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Funda\u00c3\u00a7\u00c3\u00a3o para a Ci\u00c3\u00aancia e a Tecnologia","doi-asserted-by":"publisher","award":["PTDC\/MAT-CAL\/4334\/2014"],"award-info":[{"award-number":["PTDC\/MAT-CAL\/4334\/2014"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We prove upper and lower bounds for a class of Jaeger integrals\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper G Subscript nu Baseline left-parenthesis tau right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">G<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {G}_{\\nu }(\\tau )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    appearing in axisymmetric diffusive transport related to several physical applications. In particular, we show that these integrals are globaly bounded either from above or from below by the first terms in their corresponding asymptotic expansions in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"tau\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03c4\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tau<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , both at zero and infinity. In the case of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper G 0 left-parenthesis tau right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">G<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03c4\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {G}_{0}(\\tau )<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    we show that it is bounded from below by the Ramanujan integral.\n                  <\/p>\n                  <p>These bounds are obtained as a consequence of sharp bounds derived for the modulus and phase of Hankel functions, and for the Ramanujan integral, which we believe to be new and of independent interest, complementing the asymptotic and numerical results in the literature.<\/p>","DOI":"10.1090\/mcom\/3267","type":"journal-article","created":{"date-parts":[[2017,3,8]],"date-time":"2017-03-08T09:15:12Z","timestamp":1488964512000},"page":"289-308","source":"Crossref","is-referenced-by-count":10,"title":["Sharp bounds for the modulus and phase of Hankel functions with applications to Jaeger integrals"],"prefix":"10.1090","volume":"87","author":[{"given":"Pedro","family":"Freitas","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2017,5,31]]},"reference":[{"key":"1","unstructured":"[AR] R. A. Askey and R. Roy, Gamma function, in NIST Handbook of Mathematical Functions, edited by F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, Cambridge University Press (2010)."},{"issue":"5","key":"2","doi-asserted-by":"publisher","first-page":"1257","DOI":"10.1007\/s10910-015-0485-7","article-title":"Bounds for Jaeger integrals","volume":"53","author":"Baricz, \u00c1rp\u00e1d","year":"2015","journal-title":"J. Math. Chem.","ISSN":"https:\/\/id.crossref.org\/issn\/0259-9791","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"547","DOI":"10.1512\/iumj.1971.21.21043","article-title":"Note on an asymptotic expansion","volume":"21","author":"Bouwkamp, C. J.","year":"1971","journal-title":"Indiana Univ. Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-2518","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"[BOS] D. Britz, O. \u00d8sterby and J. Strutwolf, Reference values of the chronoamperometric response at cylindrical and capped cylindrical electrodes, Electrochimica Acta 55 (2010), 5629\u20135635.","DOI":"10.1016\/j.electacta.2010.04.096"},{"key":"5","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1512\/iumj.1970.19.19040","article-title":"An integral identity due to Ramanujan which occurs in neutron transport theory","volume":"19","author":"Dorning, J. J.","year":"1969","journal-title":"J. Math. Mech."},{"key":"6","volume-title":"Ramanujan. Twelve lectures on subjects suggested by his life and work","author":"Hardy, G. H.","year":"1940"},{"key":"7","doi-asserted-by":"publisher","first-page":"154","DOI":"10.2307\/2372726","article-title":"On differential equations and the function \ud835\udc3d_{\ud835\udf07}\u00b2+\ud835\udc4c_{\ud835\udf07}\u00b2","volume":"83","author":"Hartman, Philip","year":"1961","journal-title":"Amer. J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9327","issn-type":"print"},{"key":"8","first-page":"223","article-title":"Heat flow in the region bounded internally by a circular cylinder","volume":"61","author":"Jaeger, J. C.","year":"1942","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A","ISSN":"https:\/\/id.crossref.org\/issn\/0308-2105","issn-type":"print"},{"key":"9","first-page":"229","article-title":"A short table of \u222b^{\u221e}\u2080(\ud835\udc52^{-\ud835\udc65\ud835\udc62\u00b2}\/(\ud835\udc3d\u2080\u00b2(\ud835\udc62)+\ud835\udc4c\u00b2\u2080(\ud835\udc62)))(\ud835\udc51\ud835\udc62\/\ud835\udc62)","volume":"61","author":"Jaeger, J. C.","year":"1942","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A","ISSN":"https:\/\/id.crossref.org\/issn\/0308-2105","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1017\/S0956792599004039","article-title":"The asymptotic behaviour of Ramanujan\u2019s integral and its application to two-dimensional diffusion-like equations","volume":"11","author":"Llewellyn Smith, Stefan G.","year":"2000","journal-title":"European J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0956-7925","issn-type":"print"},{"key":"11","doi-asserted-by":"crossref","unstructured":"[N] J. W. Nicholson, A problem in the theory of heat conduction, Proc. Roy. Soc. London A 100 (1921), 226\u2013240.","DOI":"10.1098\/rspa.1921.0083"},{"key":"12","unstructured":"[OM] F. W. J. Olver and L.C. Maximon, Bessel functions, in NIST Handbook of Mathematical Functions, edited by F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, Cambridge University Press (2010)."},{"issue":"2136","key":"13","doi-asserted-by":"publisher","first-page":"3570","DOI":"10.1098\/rspa.2011.0301","article-title":"On approximations to a class of Jaeger integrals","volume":"467","author":"Phillips, W. R. C.","year":"2011","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/1364-5021","issn-type":"print"},{"key":"14","series-title":"Cambridge Mathematical Library","isbn-type":"print","volume-title":"A treatise on the theory of Bessel functions","author":"Watson, G. N.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0521483913"},{"key":"15","doi-asserted-by":"publisher","first-page":"232","DOI":"10.1090\/S0002-9904-1948-08987-X","article-title":"Nicholson\u2019s integral for \ud835\udc3d_{\ud835\udc5b}\u00b2(\ud835\udc67)+\ud835\udc4c_{\ud835\udc5b}\u00b2(\ud835\udc67)","volume":"54","author":"Wilkins, J. Ernest, Jr.","year":"1948","journal-title":"Bull. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9904","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2018-87-309\/S0025-5718-2017-03267-1\/mcom3267_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"http:\/\/www.ams.org\/mcom\/2018-87-309\/S0025-5718-2017-03267-1\/S0025-5718-2017-03267-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2018-87-309\/S0025-5718-2017-03267-1\/S0025-5718-2017-03267-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:27:55Z","timestamp":1776799675000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2018-87-309\/S0025-5718-2017-03267-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5,31]]},"references-count":15,"journal-issue":{"issue":"309","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["S0025-5718-2017-03267-1"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3267","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2017,5,31]]}}}