Integrals of the Ising class

Borwein, Jonathan M. and Crandall, Richard E. and Bailey, David H. (2006) Integrals of the Ising class. J. Phys. A., 39 . pp. 12271-12302.

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    From an experimental-mathematical perspective we analyze “Isingclass” 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 had conjectured—on the basis of extreme-precision numerical quadrature—that Cn has a finite large-n limit, namely C_\infty = 2exp(−2*gamma), with gamma 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, highly parallel computation was required.

    Item Type: Article
    Additional Information: pubdom TRUE
    Subjects: 33-xx Special functions
    65-xx Numerical analysis
    82-xx Statistical mechanics, structure of matter
    Faculty: UNSPECIFIED
    Depositing User: Users 1 not found.
    Date Deposited: 05 Jun 2006
    Last Modified: 11 Jan 2015 19:32

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