- Title
- Integrals of the Ising Class
- Creator
- Bailey, D. H.; Borwein, J. M.; Crandall, R. E.
- Relation
- Journal of Physics A: Mathematical and Theoretical Vol. 39, Issue 40, p. 2271-2302
- Publisher Link
- http://dx.doi.org/10.1088/0305-4470/39/40/001
- Publisher
- Institute of Physics (IOP) Publishing
- Resource Type
- journal article
- Date
- 2006
- Description
- From an experimental-mathematical perspective we analyse 'Ising-class' integrals. These are structurally related n-dimensional integrals we call Cn, Dn, En, where Dn is a magnetic susceptibility integral central to the Ising theory of solid-state physics. We first analyse We had conjectured—on the basis of extreme-precision numerical quadrature—that Cn 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 Dn and En both decay exponentially with n, in a certain rigorous sense. While Cn, Dn 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.
- Subject
- Ising theory; Euler constant; integrals; parallel computation
- Identifier
- http://hdl.handle.net/1959.13/803609
- Identifier
- uon:6447
- Identifier
- ISSN:1751-8113
- Rights
- This is an author-created, un-copyedited version of an article accepted for publication in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The definitive publisher authenticated version is available online at http://dx.doi.org/10.1088/0305-4470/39/40/001
- Language
- eng
- Full Text
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