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Normal modes for Green's functions and the method of theta functions

Published online by Cambridge University Press:  24 October 2008

Riho Terras
Riho Terras
Affiliation:
University of California, La Jolla, California
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Abstract

The eigenfunctions for four types of bounded plane domains are constructed by a group-theoretic method. The group theory is then used to prove a completeness theorem and to derive a theta function transformation formula. Rapidly convergent series for the Green's functions of the Laplace operator on these domains are then constructed. Final formulas are summarized without reference to group theory.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 89 , Issue 2 , March 1981 , pp. 331 - 341
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

(1)Abramowitz, M. and Stegun, I.Handbook of mathematical functions (Dover, New York, 1967).Google Scholar
(2)Bertaut, F.L'Énergie électrostatiquede réseaux ioniques. J. de Physique et Rad. 13 (1952), 499–505.CrossRefGoogle Scholar
(3)Born, M. and Huang, K.Dynamical theory of crystal lattices (Clarendon Press, 1956).Google Scholar
(4)Born, M. and Misra, R. D.On the stability of crystal lattices. Proc. Cambridge Philos. Soc. 36 (1940), 466–478.CrossRefGoogle Scholar
(5)Bradburn, M.The thermodynamics of crystal lattices. Proc. Cambridge Philos. Soc. 39 (1942), 113–127.CrossRefGoogle Scholar
(6)Emersleben, O.Zetafunctionen und elektrostatische Gitterpotentiale. Phys. Zeit. 24 (1923), 73–80.Google Scholar
(7)Ewald, P. P.Zur Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. d. Phys. 64 (1921), 253–287.CrossRefGoogle Scholar
(8)Henry, N. F. M. and Lonsdale, K.International tables for X-ray crystallography i (Kynoch, Birmingham, 1976).Google Scholar
(9)Kornfeld, H.Die Berechnung elektrostatischer Potentiale und der Energie von Dipol und Quadrupolgittern. Zeit. Phys. 22 (1924), 27–43.CrossRefGoogle Scholar
(10)Lamée;, G.Leçons sur la théorie analytique de la chaleur (Paris, 1961).Google Scholar
(11)Misra, R. D.On the stability of crystal lattices. Proc. Cambridge Philos. Soc. 36 (1940), 173–182.CrossRefGoogle Scholar
(12)Morse, P. and Feshbach, H.Methods of mathematical physics, vols. I and II (Addison Wesley, New York (1967)).Google Scholar
(13)Mijboer, B. R. A. and de Wette, F. W.On the calculation of lattice sums. Physics 23 (1957), 309–321.Google Scholar
(14)Nooney, C. G.On the vibrations of triangular membranes (Dissertation, Stanford University, 1953).Google Scholar
(15)Terras, R.The determination of incomplete gamma functions. J. Computational Phys. 31 (1979), 146–151.CrossRefGoogle Scholar
(16)Terras, R.A Miller algorithm for an incomplete Bessel function. J. Computational Phys. 38 (1980). (To appear.)Google Scholar
(17)Terras, R. and Swanson, R.The solution of electrostatic image problems. Amer. J. Phys. 48 (1980), 526–531.CrossRefGoogle Scholar
(18)Terras, R. and Swanson, R., Image methods for constructing eigenfunctions and Green's functions for domains bounded by planes. J. Mathematical Phys. 21 (1980) 2140–2153.CrossRefGoogle Scholar