A Family of Berndt-Type Integrals and Associated Barnes Multiple Zeta Functions
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| Format: | Preprint |
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2025
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| _version_ | 1866918320240852992 |
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| author | Gu, Xinyue Xu, Ce Zhou, Jianing |
| author_facet | Gu, Xinyue Xu, Ce Zhou, Jianing |
| contents | In this paper, we focus on calculating a specific class of Berndt integrals, which exclusively involves (hyperbolic) cosine functions. Initially, this integral is transformed into a Ramanujan-type hyperbolic (infinite) sum via contour integration. Subsequently, a function incorporating theta is defined. By employing the residue theorem, the mixed Ramanujan-type hyperbolic (infinite) sum with both hyperbolic cosine and hyperbolic sine in the denominator is converted into a simpler Ramanujan-type hyperbolic (infinite) sum, which contains only hyperbolic cosine or hyperbolic sine in the denominator. The simpler Ramanujan-type hyperbolic (infinite) sum is then evaluated using Jacobi elliptic functions, Fourier series expansions, and Maclaurin series expansions. Ultimately, the result is expressed as a rational polynomial of Gamma and \sqrt{pi}.Additionally, the integral is related to the Barnes multiple zeta function, which provides an alternative method for its calculation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_20074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Family of Berndt-Type Integrals and Associated Barnes Multiple Zeta Functions Gu, Xinyue Xu, Ce Zhou, Jianing Mathematical Physics Number Theory 33E05, 33E20, 44A05, 11M99 F.2.2; I.2.7 In this paper, we focus on calculating a specific class of Berndt integrals, which exclusively involves (hyperbolic) cosine functions. Initially, this integral is transformed into a Ramanujan-type hyperbolic (infinite) sum via contour integration. Subsequently, a function incorporating theta is defined. By employing the residue theorem, the mixed Ramanujan-type hyperbolic (infinite) sum with both hyperbolic cosine and hyperbolic sine in the denominator is converted into a simpler Ramanujan-type hyperbolic (infinite) sum, which contains only hyperbolic cosine or hyperbolic sine in the denominator. The simpler Ramanujan-type hyperbolic (infinite) sum is then evaluated using Jacobi elliptic functions, Fourier series expansions, and Maclaurin series expansions. Ultimately, the result is expressed as a rational polynomial of Gamma and \sqrt{pi}.Additionally, the integral is related to the Barnes multiple zeta function, which provides an alternative method for its calculation. |
| title | A Family of Berndt-Type Integrals and Associated Barnes Multiple Zeta Functions |
| topic | Mathematical Physics Number Theory 33E05, 33E20, 44A05, 11M99 F.2.2; I.2.7 |
| url | https://arxiv.org/abs/2506.20074 |