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

References

  • 1. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9 Eds., Dover, New York, 1982.
  • 2. D. H. Bailey, J. M. Borwein, N. J. Calkin, et al. Experimental Mathematics in Action, A K Peters, Wellesley, MA, 2007.
  • 3. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series and Products, 6 Eds., Academic Press, USA, 2000.
  • 4. R. Reynolds, A. Stauffer, A method for evaluating definite integrals in terms of special functions with examples, Int. Math. Forum, 15 (2020), 235-244.    
  • 5. K. S. Kölbig, The polygamma function $\psi^{(k)}(x)$ for $x=\frac{1}{4}$ and $x=\frac{3}{4}$, J. Comput. Appl. Math., 75 (1996), 43-46.
  • 6. K. S. Kölbig, The polygamma function and the derivatives of the cotangent function for rational arguments, Mathematical Physics and Mathematics, 1996, CERN-CN-96-005.
  • 7. W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3 Eds., Springer-Verlag, Berlin, Heidelberg, 1966.
  • 8. J. C. McDowell, The light from population III stars, Mon. Not. R. astr. Soc., 223 (1986), 763-786.    
  • 9. W. L. Grosshandler, A. T. Modak, Radiation from nonhomogeneous combustion products, Symposium (International) on Combustion, 18 (1981), 601-609.    
  • 10. G. E. Andrews, R. Askey, R. Roy, Special Functions, Cambridge University Press, 1999.
  • 11. O. Espinosa, V. H. Moll, A generalized polygamma function, Integr. Transf. Spec. Funct. 15 (2004), 101-115.    

 

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