8 Selected References 539Many geometrical properties of a lattice are reflected in its theta function. However, a
lattice is not uniquely determined by its theta function, since there are lattices inR^4
(and in higher dimensions) which are not isometric but have the same theta function.
For applications of elliptic functions and theta functions to classical mechanics,
conformal mapping, geometry, theoretical chemistry, statistical mechanics and approxi-
mation theory, see Halphen [15] (vol. 2), Kober [19], Boset al.[7], Glasser and
Zucker [14], Baxter [5] and Todd [28]. Applications to number theory will be
considered in the next chapter.
8 SelectedReferences
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2nd ed., Springer, Berlin, 1971.
[9] D.A. Cox, The arithmetic-geometric mean of Gauss,Enseign. Math. 30 (1984), 275–330.
[10] L. Euler,Opera omnia, Ser. I, Vol. XX (ed. A. Krazer), Leipzig, 1912.
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(1885/6), 197–200.
[12] R. Fricke,Elliptische Funktionen, Encyklop ̈adie der Mathematischen Wissenschaften,
Band II, Teil 2, pp. 177–348, Teubner, Leipzig, 1921.
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[14] M.L. Glasser and I.J. Zucker, Lattice sums,Theoretical chemistry: Advances and
perspectives 5 (1980), 67–139.
[15] G.H. Halphen,Trait ́e des fonctions elliptiques et de leurs applications, 3vols.,
Gauthier-Villars, Paris, 1886–1891.
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modular forms,Adv. in Math. 53 (1984), 125–264.
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[19] H. Kober,Dictionary of conformal representations, Dover, New York, 1952.
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Math.-Verein. 18 (1909), 44–75.
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G. Olms, Hildesheim, 1973]
[22] A.M. Legendre,Trait ́e des fonctions elliptiques et des int ́egrales Eul ́eriennes, avec des
tables pour enfaciliter le calcul num ́erique, Paris, t.1 (1825), t.2 (1826), t.3 (1828).
[Microform, Readex Microprint Corporation, New York, 1970]