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A note on the Moll–Arias de Reyna integral

  • M. L. GlasserEmail author
Article

Abstract

The Moll–Arias de Reyna integral
$$\begin{aligned} \int _0^{\infty }\frac{\mathrm{d}x}{(x^2+1)^{3/2}}\frac{1}{\sqrt{\varphi (x)+\sqrt{\varphi (x)}}} \quad \text { where } \varphi (x)=1+\frac{4}{3}\left( \frac{x}{x^2+1}\right) ^2 \end{aligned}$$
is generalized and several values are given.

Keywords

Definite integral Elliptic integral Elliptic modulus Algebraic integrand 

Mathematics Subject Classification

Primary 33E05 33E20 

Notes

Acknowledgements

The author thanks Victor Moll for informing him of [2, 3].

References

  1. 1.
    Amdeberhan, T., Moll, V.: The integrals in Gradshteyn and Ryzhik. Part 14: an elementary evaluation of entry 3.411.5. Scientia 19, 97–103 (2010)MathSciNetzbMATHGoogle Scholar
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    Arias-de Reyna, J.: True value of an integral in Gradshteyn and Ryzhik’s table (2018). arxiv:1801.0964v1 [math.CA]
  3. 3.
    Blaschke, P.: Hypergeometric form of fundamental theorem of Calculus (2018). arxiv:1808.04837v1 [math.CA]
  4. 4.
    Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edn. Springer, Berlin (1971)CrossRefGoogle Scholar
  5. 5.
    Gradshteyn, I.S., Ryzhik, I.M.: In: Jeffrey, A., Zwillinger, D. (eds.) Table of Integrals, Series, and Products, 6th edn. Academic Press, New York (2000)zbMATHGoogle Scholar
  6. 6.
    Gradshteyn, I.S., Ryzhik, I.M.: In: Zwillinger, D., Moll, V. (eds.) Table of Integrals, Series, and Products, 8th edn. Academic Press, New York (2015)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Física Teórica, Atómica y Óptica, Facultad de CienciasUniversity of ValladolidValladolidSpain
  2. 2.Department of PhysicsClarkson UniversityPotsdamUSA

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