{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:53:03Z","timestamp":1776837183406,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"344","license":[{"start":{"date-parts":[[2024,7,11]],"date-time":"2024-07-11T00:00:00Z","timestamp":1720656000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12101506"],"award-info":[{"award-number":["12101506"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["21KJB110032"],"award-info":[{"award-number":["21KJB110032"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["RDF-20-01-12"],"award-info":[{"award-number":["RDF-20-01-12"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013280","name":"Major Basic Research Project of the Natural Science Foundation of the Jiangsu Higher Education Institutions","doi-asserted-by":"publisher","award":["12101506"],"award-info":[{"award-number":["12101506"]}],"id":[{"id":"10.13039\/501100013280","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013280","name":"Major Basic Research Project of the Natural Science Foundation of the Jiangsu Higher Education Institutions","doi-asserted-by":"publisher","award":["21KJB110032"],"award-info":[{"award-number":["21KJB110032"]}],"id":[{"id":"10.13039\/501100013280","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013280","name":"Major Basic Research Project of the Natural Science Foundation of the Jiangsu Higher Education Institutions","doi-asserted-by":"publisher","award":["RDF-20-01-12"],"award-info":[{"award-number":["RDF-20-01-12"]}],"id":[{"id":"10.13039\/501100013280","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006683","name":"Xi\u2019an Jiaotong-Liverpool University","doi-asserted-by":"publisher","award":["12101506"],"award-info":[{"award-number":["12101506"]}],"id":[{"id":"10.13039\/501100006683","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006683","name":"Xi\u2019an Jiaotong-Liverpool University","doi-asserted-by":"publisher","award":["21KJB110032"],"award-info":[{"award-number":["21KJB110032"]}],"id":[{"id":"10.13039\/501100006683","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100006683","name":"Xi\u2019an Jiaotong-Liverpool University","doi-asserted-by":"publisher","award":["RDF-20-01-12"],"award-info":[{"award-number":["RDF-20-01-12"]}],"id":[{"id":"10.13039\/501100006683","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We find two series expansions for Legendre\u2019s second incomplete elliptic integral\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E left-parenthesis lamda comma k right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E(\\lambda , k)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    in terms of recursively computed elementary functions. Both expansions converge at every point of the unit square in the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis lamda comma k right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(\\lambda , k)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    plane. Partial sums of the proposed expansions form a sequence of approximations to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E left-parenthesis lamda comma k right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E(\\lambda ,k)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    which are asymptotic when\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"lamda\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03bb\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\lambda<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\/or\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    tend to unity, including when both approach the logarithmic singularity\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"lamda equals k equals 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\lambda =k=1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    from any direction. Explicit two-sided error bounds are given at each approximation order. These bounds yield a sequence of increasingly precise asymptotically correct two-sided inequalities for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E left-parenthesis lamda comma k right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>\n                              \u03bb\n                              \n                            <\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E(\\lambda , k)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . For the reader\u2019s convenience we further present explicit expressions for low-order approximations and numerical examples to illustrate their accuracy. Our derivations are based on series rearrangements, hypergeometric summation algorithms and extensive use of the properties of the generalized hypergeometric functions including some recent inequalities.\n                  <\/p>","DOI":"10.1090\/mcom\/3874","type":"journal-article","created":{"date-parts":[[2023,6,14]],"date-time":"2023-06-14T09:33:35Z","timestamp":1686735215000},"page":"2769-2794","source":"Crossref","is-referenced-by-count":2,"title":["Convergent expansions and bounds for the incomplete elliptic integral of the second kind near the logarithmic singularity"],"prefix":"10.1090","volume":"92","author":[{"given":"Dmitrii","family":"Karp","sequence":"first","affiliation":[]},{"given":"Yi","family":"Zhang","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,7,11]]},"reference":[{"key":"1","unstructured":"Bille C. 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