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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
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References:

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