Research article

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

  • 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.

    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

    Related Papers:

  • 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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    [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. doi: 10.12988/imf.2020.91272
    [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. doi: 10.1093/mnras/223.4.763
    [9] W. L. Grosshandler, A. T. Modak, Radiation from nonhomogeneous combustion products, Symposium (International) on Combustion, 18 (1981), 601-609. doi: 10.1016/S0082-0784(81)80065-3
    [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. doi: 10.1080/10652460310001600573
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