Note on an integral by Anatolii Prudnikov

Robert Reynolds; Allan Stauffer; Robert Reynolds; Allan Stauffer

doi:10.3934/math.2021162

AIMS Mathematics

2021, Volume 6, Issue 3: 2680-2689. doi: 10.3934/math.2021162

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Research article

Note on an integral by Anatolii Prudnikov

Robert Reynolds ^,,
Allan Stauffer

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

Received: 17 November 2020 Accepted: 30 December 2020 Published: 04 January 2021
MSC : 01A55, 11M06, 11M35, 30-02, 30D10, 30D30, 30E20

Closed expressions for the integral

$ \begin{equation*} \int_{0}^{\infty}\frac{x^{m-1} \log ^k(a x)}{\left(x^{2 u}+1\right) \left(x^{3 u}+1\right)}dx \end{equation*} $

are given where the variables $ a $, $ k $, $ m $ and $ u $ are general complex numbers. Some of these closed expressions are given in ^[4]. Some special cases of the integral are derived and discussed.
- entries of Prudnikov,
- Lerch,
- definite integral,
- Cauchy integral,
- Mellin transform
Citation: Robert Reynolds, Allan Stauffer. Note on an integral by Anatolii Prudnikov[J]. AIMS Mathematics, 2021, 6(3): 2680-2689. doi: 10.3934/math.2021162

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Abstract

Closed expressions for the integral

$ \begin{equation*} \int_{0}^{\infty}\frac{x^{m-1} \log ^k(a x)}{\left(x^{2 u}+1\right) \left(x^{3 u}+1\right)}dx \end{equation*} $

are given where the variables $ a $, $ k $, $ m $ and $ u $ are general complex numbers. Some of these closed expressions are given in ^[4]. Some special cases of the integral are derived and discussed.

References

[1]	Yu. A. Brychkov, O. I. Marichev, N. V. Savischenko, Handbook of Mellin Transforms, CRC Press, Taylor & Francis Group, Boca Raton, Fl, 2019.
[2]	M. Abramowitz, I. A. Stegun, (Eds), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, New York, Dover, 1982.
[3]	I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 6 Ed, Academic Press, USA, 2000.
[4]	A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series, More Special Functions, USSR Academy of Sciences, Vol. 1, Moscow, 1990.
[5]	R. Reynolds, A. Stauffer, Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series, Mathematics, 7 (2019), 1099. doi: 10.3390/math7111099
[6]	R. Reynolds, A. Stauffer, A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function, Mathematics, 7 (2019), 1148. doi: 10.3390/math7121148
[7]	R. Reynolds, A. Stauffer, Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions, Mathematics, 8 (2020), 687. doi: 10.3390/math8050687
[8]	R. Reynolds, A. Stauffer, Definite integrals involving product of logarithmic functions and logarithm of square root functions expressed in terms of special functions, AIMS Mathematics, 5 (2020), 5724–5733. doi: 10.3934/math.2020367
[9]	R. Reynolds, A. Stauffer, A Method for Evaluating Definite Integrals in Terms of Special Functions with Examples, International Mathematical Forum, 15 (2020), 235–244. doi: 10.12988/imf.2020.91272
[10]	R. Reynolds, A. Stauffer, Integrals in Gradshteyn and Ryzhik: hyperbolic and algebraic functions, International Mathematical Forum, 15 (2020), 255–263. doi: 10.12988/imf.2020.91279

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