{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:38:08Z","timestamp":1776728288247,"version":"3.51.2"},"reference-count":7,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,11,21]],"date-time":"2002-11-21T00:00:00Z","timestamp":1037836800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    This paper considers the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -point Gauss-Jacobi approximation of nonsingular integrals of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"integral Subscript negative 1 Superscript 1 Baseline mu left-parenthesis t right-parenthesis phi left-parenthesis t right-parenthesis log left-parenthesis z minus t right-parenthesis normal d t\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msubsup>\n                              <mml:mo>\n                                \u222b\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msubsup>\n                            <mml:mi>\n                              \u03bc\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>\n                              \u03d5\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>z<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>t<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"normal\">d<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mi>t<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\int _{-1}^1 \\mu (t) \\phi (t) \\log (z-t)\\, \\mathrm {d}t<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , with Jacobi weight\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"mu\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03bc\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mu<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and polynomial\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"phi\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03d5\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\phi<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and derives an estimate for the quadrature error that is asymptotic as\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n right-arrow normal infinity\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u221e\n                              \n                            <\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n \\to \\infty<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The approach follows that previously described by Donaldson and Elliott. A numerical example illustrating the accuracy of the asymptotic estimate is presented. The extension of the theory to similar integrals defined on more general analytic arcs is outlined.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01366-7","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"717-727","source":"Crossref","is-referenced-by-count":3,"title":["Asymptotic estimation of Gaussian quadrature error for a nonsingular integral in potential theory"],"prefix":"10.1090","volume":"71","author":[{"given":"David","family":"Hough","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,11,21]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"573","DOI":"10.1137\/0709051","article-title":"A unified approach to quadrature rules with asymptotic estimates of their remainders","volume":"9","author":"Donaldson, J. D.","year":"1972","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"2","doi-asserted-by":"publisher","first-page":"309","DOI":"10.2307\/2004926","article-title":"Uniform asymptotic expansions of the Jacobi polynomials and an associated function","volume":"25","author":"Elliott, David","year":"1971","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","doi-asserted-by":"publisher","first-page":"221","DOI":"10.2307\/2004418","article-title":"Calculation of Gauss quadrature rules","volume":"23","author":"Golub, Gene H.","year":"1969","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","isbn-type":"print","volume-title":"Applied and computational complex analysis. Vol. 2","author":"Henrici, Peter","year":"1977","ISBN":"https:\/\/id.crossref.org\/isbn\/0471015253"},{"key":"5","isbn-type":"print","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/BFb0087897","article-title":"Conformal mapping and Fourier-Jacobi approximations","author":"Hough, David M.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/3540527680"},{"key":"6","unstructured":"[Lev91] J. Levesley, A study of Chebyshev weighted approximations to the solution of Symm\u2019s integral equation for numerical conformal mapping, Ph.D. thesis, Dept of Mathematics, Coventry Polytechnic, Coventry CV1 5FB, UK, 1991."},{"key":"7","series-title":"American Mathematical Society Colloquium Publications, Vol. XXIII","volume-title":"Orthogonal polynomials","author":"Szeg\u0151, G\u00e1bor","year":"1975","edition":"4"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01366-7\/S0025-5718-01-01366-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01366-7\/S0025-5718-01-01366-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:56:45Z","timestamp":1776725805000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01366-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,11,21]]},"references-count":7,"journal-issue":{"issue":"238","published-print":{"date-parts":[[2002,4]]}},"alternative-id":["S0025-5718-01-01366-7"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01366-7","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":[[2001,11,21]]}}}