Resurgence in the Transition Region: The Incomplete Gamma Function
We study the resurgence properties of the coefficients ₙ() appearing in the asymptotic expansion of the incomplete gamma function within the transition region. Our findings reveal that the asymptotic behaviour of ₙ() as → +∞ depends on the parity of . Both ₂ₙ₋₁() and ₂ₙ(τ) exhibit behaviours charac...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2024 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2024
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212169 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Resurgence in the Transition Region: The Incomplete Gamma Function. Gergő Nemes. SIGMA 20 (2024), 026, 14 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862709526148415488 |
|---|---|
| author | Nemes, Gergő |
| author_facet | Nemes, Gergő |
| citation_txt | Resurgence in the Transition Region: The Incomplete Gamma Function. Gergő Nemes. SIGMA 20 (2024), 026, 14 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We study the resurgence properties of the coefficients ₙ() appearing in the asymptotic expansion of the incomplete gamma function within the transition region. Our findings reveal that the asymptotic behaviour of ₙ() as → +∞ depends on the parity of . Both ₂ₙ₋₁() and ₂ₙ(τ) exhibit behaviours characterised by a leading term accompanied by an inverse factorial series, where the coefficients are once again ₂ₖ₋₁() and ₂ₖ(), respectively. Our derivation employs elementary tools and relies on the known resurgence properties of the asymptotic expansion of the gamma function and the uniform asymptotic expansion of the incomplete gamma function. To the best of our knowledge, before this paper, there has been no investigation in the existing literature regarding the resurgence properties of asymptotic expansions in transition regions.
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| first_indexed | 2026-03-19T09:56:58Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212169 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T09:56:58Z |
| publishDate | 2024 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Nemes, Gergő 2026-01-30T08:17:10Z 2024 Resurgence in the Transition Region: The Incomplete Gamma Function. Gergő Nemes. SIGMA 20 (2024), 026, 14 pages 1815-0659 2020 Mathematics Subject Classification: 34E05; 33B20 arXiv:2401.16671 https://nasplib.isofts.kiev.ua/handle/123456789/212169 https://doi.org/10.3842/SIGMA.2024.026 We study the resurgence properties of the coefficients ₙ() appearing in the asymptotic expansion of the incomplete gamma function within the transition region. Our findings reveal that the asymptotic behaviour of ₙ() as → +∞ depends on the parity of . Both ₂ₙ₋₁() and ₂ₙ(τ) exhibit behaviours characterised by a leading term accompanied by an inverse factorial series, where the coefficients are once again ₂ₖ₋₁() and ₂ₖ(), respectively. Our derivation employs elementary tools and relies on the known resurgence properties of the asymptotic expansion of the gamma function and the uniform asymptotic expansion of the incomplete gamma function. To the best of our knowledge, before this paper, there has been no investigation in the existing literature regarding the resurgence properties of asymptotic expansions in transition regions. The author’s research was supported by the JSPS KAKENHI Grants No. JP21F21020 and No. 22H01146. The author is grateful to the referees for their comprehensive evaluation of the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Resurgence in the Transition Region: The Incomplete Gamma Function Article published earlier |
| spellingShingle | Resurgence in the Transition Region: The Incomplete Gamma Function Nemes, Gergő |
| title | Resurgence in the Transition Region: The Incomplete Gamma Function |
| title_full | Resurgence in the Transition Region: The Incomplete Gamma Function |
| title_fullStr | Resurgence in the Transition Region: The Incomplete Gamma Function |
| title_full_unstemmed | Resurgence in the Transition Region: The Incomplete Gamma Function |
| title_short | Resurgence in the Transition Region: The Incomplete Gamma Function |
| title_sort | resurgence in the transition region: the incomplete gamma function |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212169 |
| work_keys_str_mv | AT nemesgergo resurgenceinthetransitionregiontheincompletegammafunction |