Certain Integrals Arising from Ramanujan's Notebooks
In his third notebook, Ramanujan claims the specific. In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if logx were replaced by log²x in the first integral and logx were inserted in the...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2015 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2015
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147147 |
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| Cite this: | Certain Integrals Arising from Ramanujan's Notebooks / B.C. Berndt, A. Straub // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 5 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862647807367708672 |
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| author | Berndt, B.C. Straub, A. |
| author_facet | Berndt, B.C. Straub, A. |
| citation_txt | Certain Integrals Arising from Ramanujan's Notebooks / B.C. Berndt, A. Straub // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 5 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In his third notebook, Ramanujan claims the specific. In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if logx were replaced by log²x in the first integral and logx were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly relating large classes of infinite integrals and illustrate these by several examples.
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| first_indexed | 2025-12-01T13:58:46Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147147 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-01T13:58:46Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Berndt, B.C. Straub, A. 2019-02-13T17:39:21Z 2019-02-13T17:39:21Z 2015 Certain Integrals Arising from Ramanujan's Notebooks / B.C. Berndt, A. Straub // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 5 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 33E20 DOI:10.3842/SIGMA.2015.083 https://nasplib.isofts.kiev.ua/handle/123456789/147147 In his third notebook, Ramanujan claims the specific. In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if logx were replaced by log²x in the first integral and logx were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly relating large classes of infinite integrals and illustrate these by several examples. This paper is a contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications.
 The full collection is available at http://www.emis.de/journals/SIGMA/OPSFA2015.html.
 We wish to thank Khristo Boyadzhiev, Larry Glasser and the referees for their careful and
 helpful suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Certain Integrals Arising from Ramanujan's Notebooks Article published earlier |
| spellingShingle | Certain Integrals Arising from Ramanujan's Notebooks Berndt, B.C. Straub, A. |
| title | Certain Integrals Arising from Ramanujan's Notebooks |
| title_full | Certain Integrals Arising from Ramanujan's Notebooks |
| title_fullStr | Certain Integrals Arising from Ramanujan's Notebooks |
| title_full_unstemmed | Certain Integrals Arising from Ramanujan's Notebooks |
| title_short | Certain Integrals Arising from Ramanujan's Notebooks |
| title_sort | certain integrals arising from ramanujan's notebooks |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147147 |
| work_keys_str_mv | AT berndtbc certainintegralsarisingfromramanujansnotebooks AT strauba certainintegralsarisingfromramanujansnotebooks |