A MAGNETIC DOUBLE INTEGRAL

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 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.


Introduction
Recall that the arithmetic-geometric mean (AGM) agm(a, b) of two positive real numbers a and b is defined as a common limit of the sequences a 0 = a, a 1 , a 2 , . .where 0 dα cos 2 α + f 2 sin 2 α is a particular instance of a classical elliptic integral.The AGM sources several beautiful structures, formulae and algorithms in mathematics; it is linked with periods of elliptic curves, modular forms and the hypergeometric function.A standard reference to all these features of the AGM is the book [4] by the Borweins.
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 [1] has come up with the double integral and conjectured that Date: 27 March 2018. 1 In this note we infer a differential equation for I 2 (f ) that implies, among other things, the involution (2).
During the preparation of this work for publication we learnt of the proof of (2) by L. Glasser and Y. Zhou [6].Though the method in [6] is seemingly less laborious than the strategy below, our approach is different and reveals further structure of I 2 (f ).

Preliminaries
First observe that and hence the even function is given by the square of an AGM.Now let θ = θ f = f d df and Then from the linear differential equation satisfied by J(f ), we prove that is an even function of f .From study of the expansion of I 2 (f ) about f = 0, we infer the following.
Theorem 1.We have In spite of its technical nature, our proof of the theorem follows a standard route.The discovery of (5) and its relevance to (2) should be counted as the principal ingredients of investigation in this note.We postpone the details of proof to Section 6. Lemma 1. Relation (5) implies the transformation in (2).(5) gives R(f ) + R(g) = 0. Thus I 2 (f ) and I 2 (g) satisfy the same inhomogeneous differential equation, which has only one solution that is regular about f = 0. Hence we only need to demonstrate that I 2 (0) = I 2 (1) to conclude that (5) implies (2).
The evaluation we expand the inner integral in terms of sin α and obtain and so we are done.
Here and in the latter proof of Theorem 1 we invoked the standard notation for the Pochhammer symbol.

Numerical evaluation of the double integral
Ausserlechner provided [1] an expression for (1) as a single integral of a complete elliptic integral, thereby reducing the burden of numerical computation of the double integral.Here we give an efficient numerical method that follows from the differential equation that we have exposed.
For f < 0 and f = −1, we use (3) to map to the region f > 0. For f > 1, we use . Thus we now have f 2 ≤ ( √ 2 − 1) 2 < 0.172, irrespective of where we began on the real line.Hence the series expansion for the odd terms in converges rapidly.The expansion coefficients a n are efficiently delivered by a recursion that follows from the differential equation ( 4) with the left-hand side given by (5).We set a −1 = 0 and a 0 = 1.Then for n > 0 we use the recursion with an inhomogeneous final term from (5).
At the CM point )] and using the geometrical Hall factor G H0 = 2/3 attributed to Haeusler [7] by Ausserlechner [1, above Eq.(54c)].Then from our Eq.(3) we obtain The modular parametrisation of the AGM and the inhomogeneous differential equation ( 4), (5) give rise to modularity properties of the integral I 2 (f ).We transform from f to the nome q defined by which gives with Then the third-order inhomogeneous equation has the form with q = exp(2πiτ ).To transform the result of Theorem 1, we use and obtain the inhomogeneous term as with integer values of A(n) for n ≤ 32200, beginning with for n = 0, 1, . . ., 9, . . . .We expect the following to be true.
Conjecture 1.All the numbers A(n) occurring in the Fourier expansion (7) of the weight 4 modular form are integral.
Hence these involutions coincide and mere algebra shows that R(g It is instructive to consider Klein's j-invariant for which we know that By subtraction, we obtain a factorisation of from which we conclude that Similarly, from we obtain We know of seven cases with positive ψ(τ 3].The corresponding values of f are given by The involution gives At the fixed point, we have ψ 0 = 1 and hence f 0 = √ 2 − 1.The remaining cases are with ψ 1 from j(i) = 12 3 and ψ 2 from j( √ 2i) = 20 3 .We used j(2i) = 66 3 to factorise with ψ 3 given by the positive root of the quadratic factor.Thus we obtain from the Chowla-Selberg formula at the first singular value.At the second, we have Moreover, we obtain values for , as follows: Conjecture 1 means that the anti-derivative of φ(τ ) has a Fourier expansion with integer coefficients, a somewhat unusual property for a modular form (of weight 4).This is possible only because φ(τ ) has poles in the upper-half of the complex plane.The dominant singularity is at τ = (1 + i)/2, where ψ(τ ) = −1 and hence . This led us to expect an exponential growth of the form A(n) = (−1) n (C + O(1/n))e nπ at large n, for some positive constant C. Empirically, we discovered that C = 1 2 e π/2 and even more remarkably that ) with a very simple exponentially increasing function Thus the leading term gives 80% of the decimal digits of A(n).Intrigued by this, we studied the next to leading term, finding ), which determines more than 92% of the digits.This led us to suppose that with trigonometrical coefficients C(m, n) that are non-zero if and only if m belongs to a sequence S of integers, beginning with 1, 5, 13.Further numerical work revealed that this sequence continues as follows: 1, 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, . . ., from which we inferred that S is the sequence of positive integers divisible only by primes congruent to 1 modulo 4.
k with distinct primes p k ≡ 1 mod 4, there is a positive integer r(m) < m/2 such that (8) holds with Furthermore, C(m, n) = 0 for m > 1 not of the latter form.
For ω = 1 and m ∈ S, a unique positive integer r(m) < m/2 specifies C(m, n).For ω > 1, there are 2 ω−1 such integers, of which we select the smallest.For example r( 65 This fit gave 99.9% of the digits of A(n).We continued up to m = 2017 and obtained this fit gave 99.95% of the digits of A(n).Then we discovered an algebraic method.To find r(m), we may study the orbit of (1 + i)/2 in Γ 0 (2), which is the group of Möbius transformations M(a, b, c, d) with integer arguments such that ad − bc = 1 and, most crucially, c ≡ 0 mod 2. The action is specified by τ → (aτ + b)/(cτ + d).Since c is even, a and d must be odd.Acting on the singularity of φ(τ ) at τ = (1 + i)/2, the transformation M(a, b, c, d) locates another singularity at There are 2 ω−1 essentially different ways of expressing an integer m ∈ S as a sum of coprime squares.Suppose that we take one of these, say m = γ 2 +δ 2 with positive odd γ.Then we set c = 2γ and d = δ − γ.For a, we take the unique positive odd integer with a < c and c | (ad − 1).Then we set b = (ad − 1)/c and compute w by the formula above.For precisely one of the two choices of sign for δ we obtain an odd positive integer w < m.If ω = 1, we finish by setting r(m) = (m − w)/2.
After more algebraic investigation, we were led to posit that Thereafter, algebra becomes tedious.However, a numerical method is efficient, thanks to fast evaluation of eta values by Pari-GP, which took merely a minute to evaluate 420 000 digits of D and φ(τ + + 10 −510 i).Then rational values of c n for n ∈ [1,200] followed in seconds.This data determines 804 terms in the Laurent expansion.The following properties were observed: (a) c n = a n /b n is a positive ratio of odd integers; (b) no prime greater than 4n + 1 divides b n ; (c) the denominator of (4n + 5)! c n is square free and divisible only by primes p < n with p ≡ 1 mod 4. At large n, with a leading term from the singularity at τ = (3 + i)/10 and an oscillating next to leading term from the singularity at τ = (5 + i)/26.We used the sum rule to test our results.This follows from the vanishing of (10) at τ = i/ √ 2. Using exact values of c n for n ≤ 200 and thereafter the approximation c n D n ≈ 8n − 6, we obtained a left hand side differing from unity by less than 10 −265 .

The inhomogeneous differential equation
In this section we prove Theorem 1. Write the double integral (1) as , ), and perform the change .

Consider the series
where the standard hypergeometric notation is used [2] and a transformation due to Thomae [2, Sect.3.2] is applied.
The representation (11) can be combined with the classical Gosper-Zeilberger algorithm to produce a differential equation for Y (h).
Theorem 2. In a neighbourhood of h = 0 the function Y (h) satisfies where .
Applying (14) this finally leads to which does not involve sums S 2 (n).Notice that T (n) = n m=1 c(n; m), where Summing the equality over m = 1, . . ., n we have The recursion immediately implies that and also simplifies the right-hand side of (15): to translate the resulting difference equation into the differential equation (12).
It is clear that the singularities of the differential equation ( 12) are at the points h = 0, 1, 2, ∞, where the singularity at h = 2 occurs because of the inhomogeneity.
Proof of Theorem 1.We can translate the differential equation of Theorem 2 into one for the function using the defining expression The result is 1 It follows then from Lemma 1 that In summary, our proof exploits a power-series development of (1) in a suitable base (namely, with respect to h = 4f /(1 + f ) 2 ) and its algorithmic fitness for a telescoping argument.This structure suggests the existence of double integrals that are similar to or more general than Ausserlechner's integral, which satisfy arithmetic inhomogeneous differential equations and have transformation properties of type (2).

Final remarks
A different, two-variable generalization of the complete elliptic integral was considered by W. N. Bailey in [3].
A general form of the Taylor expansion of a modular form near a CM (elliptic) point (that is, a quadratic irrationality from the upper-half of the complex plane) is analysed in [8].The principal theorem there can be used to explain the particular form of Laurent expansion in (10).
The differential equation for (1) reveals an inhomogeneous term (5) with the three symmetries The last of these is derivable from, yet obscured by, the expansion in h = 4f /(1+f ) 2 .Finally, we point out again the mysteries behind a modular parametrisation of the double integral: our unexpected discovery of the integrality of the sequence A(0), A(1), A(2), . . . in Section 3 (Conjecture 1) and an arithmetic pattern of the asymptotic growth for the sequence in Section 4 (Conjecture 2).