Please use this identifier to cite or link to this item: http://hdl.handle.net/1959.13/803609
- Integrals of the Ising Class
Bailey, D. H.;
Borwein, J. M.;
Crandall, R. E.
- 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.
- Journal of Physics A: Mathematical and Theoretical Vol. 39, Issue 40, p. 2271-2302
- Publisher Link
- Institute of Physics (IOP) Publishing
- Resource Type
- journal article