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# On The Simple Cubic Lattice Green Function

G. S. Joyce
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 273, No. 1236 (Feb. 8, 1973), pp. 583-610
Stable URL: http://www.jstor.org/stable/74037
Page Count: 28
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## Abstract

The analytical properties of the simple cubic lattice Green function G(t) = $\frac{1}{\pi ^{3}}\mathop{\iiint}_{0}^{\pi}$[t-(cos x1 + cos x2 + cos x3)]-1dx1dx$_{2}$dx3 are investigated. In particular, it is shown that tG(t) can be written in the form tG(t) = [F(9, -3/4; 1/4, 3/4, 1, 1/2; 9/t2)]2, where F(a, b; α , β , γ , δ ; z) denotes a Heun function. The standard analytic continuation formulae for Heun functions are then used to derive various expansions for the Green function G-(s) $\equiv$ GR(s) + iGI(s) = $\underset \epsilon \rightarrow 0+\to{\lim}$ G(s - iε ) (0 ≤ s < ∞ ) about the points s = 0, 1 and 3. From these expansions accurate numerical values of GR(s) and GI(s) are obtained in the range 0 ≤ s ≤ 3, and certain new summation formulae for Heun functions of unit argument are deduced. Quadratic transformation formulae for the Green function G(t) are discussed, and a connexion between G(t) and the Lamé-Wangerin differential equation is established. It is also proved that G(t) can be expressed as a product of two complete elliptic integrals of the first kind. Finally, several applications of the results are made in lattice statistics.

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