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

Robert Reynolds; Allan Stauffer; Robert Reynolds; Allan Stauffer

doi:10.3934/math.2020463

AIMS Mathematics

2020, Volume 5, Issue 6: 7252-7258. doi: 10.3934/math.2020463

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Research article

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

Robert Reynolds ^,,
Allan Stauffer

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

Received: 11 July 2020 Accepted: 14 September 2020 Published: 16 September 2020
MSC : 01A55, 11M06, 11M35, 30-02, 30D10, 30D30, 30E20

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.
- Binet,
- log gamma,
- Planck’s Law,
- Euler’s constant,
- Catalan’s constant,
- logarithmic function,
- definite integral,
- Hexagamma,
- Cauchy integral,
- entries of Gradshteyn and Ryzhik
Citation: Robert Reynolds, Allan Stauffer. Derivation of logarithmic integrals expressed in teams of the Hurwitz zeta function[J]. AIMS Mathematics, 2020, 5(6): 7252-7258. doi: 10.3934/math.2020463

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Abstract

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.

References

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