{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:52:01Z","timestamp":1776721921905,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"215","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    A new asymptotic expansion is derived for the incomplete beta function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper I left-parenthesis a comma b comma x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">I(a,b,x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , which is suitable for large\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , small\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b\">\n                        <mml:semantics>\n                          <mml:mi>b<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">b<\/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=\"x greater-than 0.5\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0.5<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x &gt; 0.5<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This expansion is of the form\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper I left-parenthesis a comma b comma x right-parenthesis tilde upper Q left-parenthesis b comma minus gamma log x right-parenthesis plus StartFraction normal upper Gamma left-parenthesis a plus b right-parenthesis Over normal upper Gamma left-parenthesis a right-parenthesis normal upper Gamma left-parenthesis b right-parenthesis EndFraction x Superscript gamma Baseline sigma-summation Underscript n equals 0 Overscript normal infinity Endscripts upper T Subscript n Baseline left-parenthesis b comma x right-parenthesis slash gamma Superscript n plus 1 Baseline comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>I<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mspace width=\"1em\"\/>\n                            <mml:mo>\n                              \u223c\n                              \n                            <\/mml:mo>\n                            <mml:mspace width=\"1em\"\/>\n                            <mml:mi>Q<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03b3\n                              \n                            <\/mml:mi>\n                            <mml:mi>log<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mfrac>\n                                <mml:mrow>\n                                  <mml:mi mathvariant=\"normal\">\n                                    \u0393\n                                    \n                                  <\/mml:mi>\n                                  <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                  <mml:mi>a<\/mml:mi>\n                                  <mml:mo>+<\/mml:mo>\n                                  <mml:mi>b<\/mml:mi>\n                                  <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <\/mml:mrow>\n                                <mml:mrow>\n                                  <mml:mi mathvariant=\"normal\">\n                                    \u0393\n                                    \n                                  <\/mml:mi>\n                                  <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                  <mml:mi>a<\/mml:mi>\n                                  <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <mml:mi mathvariant=\"normal\">\n                                    \u0393\n                                    \n                                  <\/mml:mi>\n                                  <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                  <mml:mi>b<\/mml:mi>\n                                  <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                <\/mml:mrow>\n                              <\/mml:mfrac>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03b3\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:munderover>\n                              <mml:mo>\n                                \u2211\n                                \n                              <\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>=<\/mml:mo>\n                                <mml:mn>0<\/mml:mn>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"normal\">\n                                  \u221e\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:munderover>\n                            <mml:msub>\n                              <mml:mi>T<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msup>\n                              <mml:mi>\n                                \u03b3\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*}I(a,b,x) \\quad \\sim \\quad Q(b, -\\gamma \\log x) + {\\frac {\\Gamma (a + b)}{\\Gamma (a) \\Gamma (b)}} x^{\\gamma } \\sum ^{\\infty }_{n=0}T_{n}(b,x)\/ \\gamma ^{n+1} , \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q\">\n                        <mml:semantics>\n                          <mml:mi>Q<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">Q<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the incomplete Gamma function ratio and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"gamma equals a plus left-parenthesis b minus 1 right-parenthesis slash 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b3\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>a<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\gamma = a + (b - 1)\/2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . This form has some advantages over previous asymptotic expansions in this region in which\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper T Subscript n\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">T_{n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    depends on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"a\">\n                        <mml:semantics>\n                          <mml:mi>a<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">a<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    as well as on\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b\">\n                        <mml:semantics>\n                          <mml:mi>b<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">b<\/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=\"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                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00729-6","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1283-1288","source":"Crossref","is-referenced-by-count":6,"title":["An asymptotic expansion for the incomplete beta function"],"prefix":"10.1090","volume":"65","author":[{"given":"B.","family":"Doman","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","unstructured":"M. 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Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"7","series-title":"Mathematics in Science and Engineering, Vol. 53","volume-title":"The special functions and their approximations, Vol. I","author":"Luke, Yudell L.","year":"1969"},{"key":"8","doi-asserted-by":"crossref","unstructured":"E. C. Molina, Expansions for Laplacian Integrals in Terms of Incomplete Gamma Functions, International Congress of Mathematicians, Zurich, Bell System Technical Journal 11 (1932), 563-575 and Monograph B704.","DOI":"10.1002\/j.1538-7305.1932.tb00640.x"},{"issue":"6","key":"9","doi-asserted-by":"publisher","first-page":"1638","DOI":"10.1137\/0518118","article-title":"Incomplete Laplace integrals: uniform asymptotic expansion with application to the incomplete beta function","volume":"18","author":"Temme, N. M.","year":"1987","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"10","doi-asserted-by":"publisher","first-page":"609","DOI":"10.2307\/1968945","article-title":"A characterization of Boolean algebras","volume":"40","author":"Birkhoff, Garrett","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"11","doi-asserted-by":"publisher","first-page":"712","DOI":"10.2307\/1968951","article-title":"Rings with minimal condition for left ideals","volume":"40","author":"Hopkins, Charles","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00729-6\/S0025-5718-96-00729-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00729-6\/S0025-5718-96-00729-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:12:57Z","timestamp":1776719577000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1996-65-215\/S0025-5718-96-00729-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996]]},"references-count":11,"journal-issue":{"issue":"215","published-print":{"date-parts":[[1996,7]]}},"alternative-id":["S0025-5718-96-00729-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-96-00729-6","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":[[1996]]}}}