## A magnetic double integral.(English)Zbl 1470.11068

Summary: In a recent study of how the output voltage of a Hall plate is affected by the shape of the plate and the size of its contacts, U. Ausserlechner [“A method to compute the Hall-geometry factor at weak magnetic field in closed analytical form”, Electr. Eng. 98, No. 3, 189–206 (2016; doi:10.1007/s00202-015-0351-4)] has come up with a remarkable double integral that can be viewed as a generalisation of the classical elliptic ’arithmetic – geometric mean (AGM)’ integral. Here we discuss transformation properties of the integral, which were experimentally observed by Ausserlechner, as well as its analytical and arithmetic features including connections with modular forms.

### MSC:

 11F11 Holomorphic modular forms of integral weight 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33E05 Elliptic functions and integrals 33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.) 82D40 Statistical mechanics of magnetic materials
Full Text:

### References:

 [1] Ausserlechner, U., A method to compute the Hall-geometry factor at weak magnetic field in closed analytical form, Electr. Eng., 98, 189-206, (2016) [2] Bailey, W. N., Generalized Hypergeometric Series, 32, (1935), Cambridge University Press: Cambridge University Press, Cambridge · JFM 61.0406.01 [3] Bailey, W. N., A double integral, J. Lond. Math. Soc., 23, 235-237, (1948) · Zbl 0033.26401 [4] Borwein, J. M.; Borwein, P. B., Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, (1987), Wiley: Wiley, New York · Zbl 0611.10001 [5] Chowla, S.; Selberg, A., On Epstein’s zeta function (I), Proc. Natl Acad. Sci. USA, 35, 371-374, (1949) · Zbl 0032.39103 [6] Glasser, M. L.; Zhou, Y., A functional identity involving elliptic integrals, Ramanujan J., 47, 243-251, (2018) · Zbl 1401.33015 [7] Haeusler, J., Exakte Lösungen von Potentialproblemen beim Halleffekt durch konforme Abbildung, Solid-State Electron., 9, 417-441, (1966) [8] Wolfart, J., Taylorentwicklungen automorpher Formen und ein Transzendenzproblem aus der Uniformisierungstheorie, Abh. Math. Semin. Univ. Hambg., 54, 25-33, (1984) · Zbl 0556.10020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.