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

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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