Lerch $Φ$ asymptotics
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929283814916096 |
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| author | Daalhuis, Adri B. Olde |
| author_facet | Daalhuis, Adri B. Olde |
| contents | We use a Mellin-Barnes integral representation for the Lerch transcendent $Φ(z,s,a)$ to obtain large $z$ asymptotic approximations. The simplest divergent asymptotic approximation terminates in the case that $s$ is an integer. For non-integer $s$ the asymptotic approximations consists of the sum of two series. The first one is in powers of $(\ln z)^{-1}$ and the second one is in powers of $z^{-1}$. Although the second series converges, it is completely hidden in the divergent tail of the first series. We use resummation and optimal truncation to make the second series visible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_11886 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lerch $Φ$ asymptotics Daalhuis, Adri B. Olde Classical Analysis and ODEs 11M35, 30E15, 41A30, 41A60 We use a Mellin-Barnes integral representation for the Lerch transcendent $Φ(z,s,a)$ to obtain large $z$ asymptotic approximations. The simplest divergent asymptotic approximation terminates in the case that $s$ is an integer. For non-integer $s$ the asymptotic approximations consists of the sum of two series. The first one is in powers of $(\ln z)^{-1}$ and the second one is in powers of $z^{-1}$. Although the second series converges, it is completely hidden in the divergent tail of the first series. We use resummation and optimal truncation to make the second series visible. |
| title | Lerch $Φ$ asymptotics |
| topic | Classical Analysis and ODEs 11M35, 30E15, 41A30, 41A60 |
| url | https://arxiv.org/abs/2311.11886 |