New closed forms for a dilogarithmic integral, related integrals, and series
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909379209461760 |
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| author | Abdulsalam, Abdulhafeez A. |
| author_facet | Abdulsalam, Abdulhafeez A. |
| contents | In this study, we present a new closed form for the generalized integral $$\int_0^1 \frac{\mathrm{Li}_2(z) \ln(1+az)}{z}\, \mathrm{d}z,$$ where $a \in \mathbb{C} \setminus(-\infty, -1)$ and $\mathrm{Li}_2(z)$ is the dilogarithm function. This generalization is achieved by leveraging our established findings in conjunction with Vălean's results. Furthermore, we provide explicit closed forms for associated integrals, prove a transformation formula for double infinite series, expressing them as the sum of the square of an infinite series and another infinite series. We utilize this relationship to derive a novel closed form for the generalized series $$\sum_{k=1}^\infty \frac{ ζ\left(m, \frac{rk-s}{r}\right) }{(rk-s)^m},$$ for $\Re(m) > 1$, $r, s \in \mathbb{C}$, where $r \neq 0$, $rk \neq s$, for any positive integer $k$, and $ζ(s, z)$ denotes the Hurwitz zeta function. Utilizing Hermite's integral representation for $ζ(s, z)$, we derive a family of integrals from this series. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_11459 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | New closed forms for a dilogarithmic integral, related integrals, and series Abdulsalam, Abdulhafeez A. Classical Analysis and ODEs 11B68, 11M35, 33B15, 33B30 In this study, we present a new closed form for the generalized integral $$\int_0^1 \frac{\mathrm{Li}_2(z) \ln(1+az)}{z}\, \mathrm{d}z,$$ where $a \in \mathbb{C} \setminus(-\infty, -1)$ and $\mathrm{Li}_2(z)$ is the dilogarithm function. This generalization is achieved by leveraging our established findings in conjunction with Vălean's results. Furthermore, we provide explicit closed forms for associated integrals, prove a transformation formula for double infinite series, expressing them as the sum of the square of an infinite series and another infinite series. We utilize this relationship to derive a novel closed form for the generalized series $$\sum_{k=1}^\infty \frac{ ζ\left(m, \frac{rk-s}{r}\right) }{(rk-s)^m},$$ for $\Re(m) > 1$, $r, s \in \mathbb{C}$, where $r \neq 0$, $rk \neq s$, for any positive integer $k$, and $ζ(s, z)$ denotes the Hurwitz zeta function. Utilizing Hermite's integral representation for $ζ(s, z)$, we derive a family of integrals from this series. |
| title | New closed forms for a dilogarithmic integral, related integrals, and series |
| topic | Classical Analysis and ODEs 11B68, 11M35, 33B15, 33B30 |
| url | https://arxiv.org/abs/2309.11459 |