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The Ramanujan Journal

, Volume 47, Issue 2, pp 243–251 | Cite as

A functional identity involving elliptic integrals

  • M. Lawrence Glasser
  • Yajun Zhou
Article
  • 208 Downloads

Abstract

We show that the following double integral
$$\begin{aligned} \int _{0}^\pi \mathrm {d}\, x\int _0^x\mathrm {d}\, y\frac{1}{\sqrt{1-\smash [b]{p}\cos x}\sqrt{1+\smash [b]{q\cos y}}} \end{aligned}$$
remains invariant as one trades the parameters p and q for \(p'=\sqrt{1-p^2}\) and \(q'=\sqrt{1-q^2}\), respectively. This invariance property is suggested from symmetry considerations in the operating characteristics of a semiconductor Hall effect device.

Keywords

Incomplete elliptic integrals Complete elliptic integrals Landen’s transformation 

Mathematics Subject Classification

33E05 (Primary) 78A35 (Secondary) 

Notes

Acknowledgements

M.L.G. thanks Udo Ausserlechner (Infineon Technologies) and Michael Milgram (Geometrics Unlimited) for insightful correspondence. We thank an anonymous referee for valuable suggestions in improving the presentation of this paper.

References

  1. 1.
    Ausserlechner, U.: Closed form expressions for sheet resistance and mobility from Van-der-Pauw measurement on \(90^\circ \) symmetric devices with four arbitrary contacts. Solid State Electron. 116, 46–55 (2016)CrossRefGoogle Scholar
  2. 2.
    Ausserlechner, U.: A method to compute the Hall-geometry factor at weak magnetic field in closed analytical form. Electr. Eng. 98, 189–206 (2016)CrossRefGoogle Scholar
  3. 3.
    Berndt, B.C.: Ramanujan’s Notebooks (Part V). Springer, New York, NY (1998)CrossRefGoogle Scholar
  4. 4.
    Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed. Grundlehren der mathematischen Wissenschaften, vol. 67. Springer, Berlin (1971)Google Scholar
  5. 5.
    Kontsevich, M., Zagier, D.: Periods. In: Enquist, B., Schmid, W. (eds.) Mathematics Unlimited—2001 and Beyond, pp. 771–808. Springer, Berlin (2001)CrossRefGoogle Scholar
  6. 6.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, volume 1: Elementary Functions. Gordon and Breach Science Publishers, New York, NY (1986) (Translated from the Russian by N. M. Queen)Google Scholar
  7. 7.
    Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, Volume 3: More Special Functions. Gordon and Breach Science Publishers, New York, NY (1990) (Translated from the Russian by G. G. Gould)Google Scholar
  8. 8.
    Zhou, Y.: Kontsevich–Zagier integrals for automorphic Green’s functions. II. Ramanujan J. 42: 623–688 (2017) (see URL: arXiv:1506.00318v3 [math.NT] for erratum/addendum)

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Dpto. de Física Teórica, Facultad de CienciasUniversidad de ValladolidValladolidSpain
  2. 2.Donostia International Physics CenterSan SebastiánSpain
  3. 3.Program in Applied and Computational Mathematics (PACM)Princeton UniversityPrincetonUSA
  4. 4.Academy of Advanced Interdisciplinary Sciences (AAIS)Peking UniversityBeijingPeople’s Republic of China

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