AIMS Mathematics, 2020, 5(6): 7252-7258. doi: 10.3934/math.2020463.

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Derivation of logarithmic integrals expressed in teams of the Hurwitz zeta function

Department of Mathematics, York University, 4700 Keele Street, Toronto, M3J1P3, Canada

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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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Citation: Robert Reynolds, Allan Stauffer. Derivation of logarithmic integrals expressed in teams of the Hurwitz zeta function. AIMS Mathematics, 2020, 5(6): 7252-7258. doi: 10.3934/math.2020463

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