Exact Values of the Gamma Function from Stirling’s Formula
In this work the complete version of Stirling’s formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma function over all branches of the complex plane. Exact values can only be obtained by regularization. Two m...
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MDPI AG
2020-07-01
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| Online Access: | https://www.mdpi.com/2227-7390/8/7/1058 |
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| author | Victor Kowalenko |
| author_facet | Victor Kowalenko |
| author_sort | Victor Kowalenko |
| collection | DOAJ |
| description | In this work the complete version of Stirling’s formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma function over all branches of the complex plane. Exact values can only be obtained by regularization. Two methods are introduced: Borel summation and Mellin–Barnes (MB) regularization. The Borel-summed remainder is composed of an infinite convergent sum of exponential integrals and discontinuous logarithmic terms that emerge in specific sectors and on lines known as Stokes sectors and lines, while the MB-regularized remainders reduce to one complex MB integral with similar logarithmic terms. As a result that the domains of convergence overlap, two MB-regularized asymptotic forms can often be used to evaluate the logarithm of the gamma function. Though the Borel-summed remainder has to be truncated, it is found that both remainders when summed with (1) the truncated asymptotic series, (2) Stirling’s formula and (3) the logarithmic terms arising from the higher branches of the complex plane yield identical values for the logarithm of the gamma function. Where possible, they also agree with results from Mathematica. |
| first_indexed | 2024-03-10T18:45:31Z |
| format | Article |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T18:45:31Z |
| publishDate | 2020-07-01 |
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| spelling | doaj.art-6e26b17ce4bb449284d22be1abba8dc12023-11-20T05:32:45ZengMDPI AGMathematics2227-73902020-07-0187105810.3390/math8071058Exact Values of the Gamma Function from Stirling’s FormulaVictor Kowalenko0School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, AustraliaIn this work the complete version of Stirling’s formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma function over all branches of the complex plane. Exact values can only be obtained by regularization. Two methods are introduced: Borel summation and Mellin–Barnes (MB) regularization. The Borel-summed remainder is composed of an infinite convergent sum of exponential integrals and discontinuous logarithmic terms that emerge in specific sectors and on lines known as Stokes sectors and lines, while the MB-regularized remainders reduce to one complex MB integral with similar logarithmic terms. As a result that the domains of convergence overlap, two MB-regularized asymptotic forms can often be used to evaluate the logarithm of the gamma function. Though the Borel-summed remainder has to be truncated, it is found that both remainders when summed with (1) the truncated asymptotic series, (2) Stirling’s formula and (3) the logarithmic terms arising from the higher branches of the complex plane yield identical values for the logarithm of the gamma function. Where possible, they also agree with results from Mathematica.https://www.mdpi.com/2227-7390/8/7/1058asymptotic seriesasymptotic formBorel summationcomplete asymptotic expansiondivergent seriesdomain of convergence |
| spellingShingle | Victor Kowalenko Exact Values of the Gamma Function from Stirling’s Formula Mathematics asymptotic series asymptotic form Borel summation complete asymptotic expansion divergent series domain of convergence |
| title | Exact Values of the Gamma Function from Stirling’s Formula |
| title_full | Exact Values of the Gamma Function from Stirling’s Formula |
| title_fullStr | Exact Values of the Gamma Function from Stirling’s Formula |
| title_full_unstemmed | Exact Values of the Gamma Function from Stirling’s Formula |
| title_short | Exact Values of the Gamma Function from Stirling’s Formula |
| title_sort | exact values of the gamma function from stirling s formula |
| topic | asymptotic series asymptotic form Borel summation complete asymptotic expansion divergent series domain of convergence |
| url | https://www.mdpi.com/2227-7390/8/7/1058 |
| work_keys_str_mv | AT victorkowalenko exactvaluesofthegammafunctionfromstirlingsformula |