Integrals of Frullani type and the method of brackets

  • 1 Departmento de Fisica, Valparaiso, Chile
  • 2 Departmento de Fisica y Astronomia, Avda. Gran Bretan a 1111, Valparaiso, Chile
  • 3 Department of Mathematics, MS 39560, Long Beach, USA
  • 4 Department of Mathematics, LA 70118, New Orleans, USA

Abstract

The method of brackets is a collection of heuristic rules, some of which have being made rigorous, that provide a flexible, direct method for the evaluation of definite integrals. The present work uses this method to establish classical formulas due to Frullani which provide values of a specific family of integrals. Some generalizations are established.

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  • [1]

    M. Albano, T. Amdeberhan, E. Beyerstedt, and V. Moll. The integrals in Gradshteyn and Ryzhik. Part 15: Frullani integrals. Scientia, 19:113-119, 2010.

  • [2]

    T. Amdeberhan, O. Espinosa, I. Gonzalez, M. Harrison, V. Moll, and A. Straub. Ramanujan Master Theorem. The Ramanujan Journal, 29:103-120, 2012.

  • [3]

    J. Arias-de Reyna. On the theorem of Frullani. Proc. Amer. Math. Soc., 109:165-175, 1990.

  • [4]

    J. M. Borwein and P. B. Borwein. Pi and the AGM- A study in analytic number theory and computational complexity. Wiley, New York, 1st edition, 1987.

  • [5]

    I. Gonzalez, K. Kohl, and V. Moll. Evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets. Scientia, 25:65-84,2014.

  • [6]

    I. Gonzalez and M. Loewe. Feynman diagrams and a combination of the Integration by Parts (IBP) and the Integration by Fractional Expansion (IBFE) Techniques. Physical Review D, 81:026003, 2010.

  • [7]

    I. Gonzalez and V. Moll. Definite integrals by the method of brackets. Part 1. Adv. Appl. Math., 45:50-73, 2010.

  • [8]

    I. Gonzalez, V. Moll, and A. Straub. The method of brackets. Part 2: Examples and applications. In T. Amdeberhan, L. Medina, and Victor H. Moll, editors, Gems in Experimental Mathematics, volume 517 of Contemporary Mathematics, pages 157-172. American Mathematical Society, 2010.

  • [9]

    I. Gonzalez and I. Schmidt. Optimized negative dimensional integration method (NDIM) and multiloop Feynman diagram calculation. Nuclear Physics B, 769:124-173, 2007.

  • [10]

    I. Gonzalez and I. Schmidt. Modular application of an integration by fractional expansion (IBFE) method to multiloop Feynman diagrams. Phys. Rev. D, 78:086003, 2008.

  • [11]

    I. Gonzalez and I. Schmidt. Modular application of an integration by fractional expansion (IBFE) method to multiloop Feynman diagrams II. Phys. Rev. D, 79:126014, 2009.

  • [12]

    I. S. Gradshteyn and I. M. Ryzhik. Table of Integrals, Series, and Products. Edited by D. Zwillinger and V. Moll. Academic Press, New York, 8th edition, 2015.

  • [13]

    G. H. Hardy. On the Frullanian integral 0 [ φ a x m φ b x n ] / x log x p d x . $\int_0^\infty {\left( {[\varphi \left( {a{x^m}} \right) - \varphi \left( {b{x^n}} \right)]/x} \right)} {\left( {\log \,x} \right)^p}\,dx.$ Quart. J. Math., 33:113-144, 1902.

  • [14]

    L. Medina and V. Moll. The integrals in Gradshteyn and Ryzhik. Part 10: The digamma function. Scientia, 17:45-66, 2009.

  • [15]

    A. M. Ostrowski. On some generalizations of the Cauchy-Frullani integral. Proc. Nat. Acad. Sci. U.S.A, 35:612-616, 1949.

  • [16]

    A. M. Ostrowski. On Cauchy-Frullani integrals. Comment. Math. Helvetici, 51:57-91, 1976.

  • [17]

    E. T. Whittaker and G. N. Watson. Modern Analysis. Cambridge University Press, 1962.

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