Integrals of the Ising class

doi:10.1088/0305-4470/39/40/001

Integrals of the Ising class

D H Bailey et al 2006 J. Phys. A: Math. Gen. 39 12271-12302 doi: 10.1088/0305-4470/39/40/001

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D H Bailey¹, J M Borwein² and R E Crandall³
¹ Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
² Faculty of Computer Science, Dalhousie University, Halifax, NS, B3H 2W5, Canada
³ Center for Advanced Computation, Reed College, Portland, OR, USA
E-mail: dhbailey@lbl.gov, jborwein@cs.dal.ca and crandall@reed.edu

Abstract. From an experimental-mathematical perspective we analyse 'Ising-class' integrals. These are structurally related n-dimensional integrals we call C_n, D_n, E_n, where D_n is a magnetic susceptibility integral central to the Ising theory of solid-state physics. We first analyse

$C_n := \frac4{n!} \int_{0}^{\infty} \cdots \int_{0}^{\infty} \frac{1} {\big(\!\sum_{j=1}^{n} (u_j + 1/u_j) \big)^2} \frac{{\rm d}u_1}{u_1} \cdots \frac{{\rm d}u_n}{u_n}.$

We had conjectured—on the basis of extreme-precision numerical quadrature—that C_n has a finite large-n limit, namely C_∞ = 2 e^−2γ, with γ being the Euler constant. On such a numerological clue we are able to prove the conjecture. We then show that integrals D_n and E_n both decay exponentially with n, in a certain rigorous sense. While C_n, D_n remain unresolved for n ≥ 5, we were able to conjecture a closed form for E₅. Our experimental results involved extreme-precision, multidimensional quadrature on intricate integrands; thus, a highly parallel computation was required.

PACS numbers: 02.60.Jh, 05.10.Ln, 05.50.+q

Mathematics Subject Classification: 65D30, 82B20, 82B80

Print publication: Issue 40 (6 October 2006)
Received 2 June 2006, in final form 14 August 2006
Published 19 September 2006

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