Derivation of logarithmic integrals expressed in teams of the Hurwitz zeta function
In this paper by means of contour integration we will evaluate definite integrals of the form \begin{equation*} \int_{0}^{1}\left(\ln^k(ay)-\ln^k\left(\frac{a}{y}\right)\right)R(y)dy \end{equation*} in terms of a special function, where $R(y)$ is a general function and $k$ and $a$ are arbitrary comp...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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AIMS Press
2020-09-01
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| Series: | AIMS Mathematics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/10.3934/math.2020463/fulltext.html |
| Summary: | In this paper by means of contour integration we will evaluate definite integrals of the form \begin{equation*} \int_{0}^{1}\left(\ln^k(ay)-\ln^k\left(\frac{a}{y}\right)\right)R(y)dy \end{equation*} in terms of a special function, where $R(y)$ is a general function and $k$ and $a$ are arbitrary complex numbers. These evaluations can be expressed in terms of famous mathematical constants such as the Euler's constant $\gamma$ and Catalan's constant $C$. Using derivatives, we will also derive new integral representations for some Polygamma functions such as the Digamma and Trigamma functions along with others. |
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| ISSN: | 2473-6988 |