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Integral Equations and the Determination of Green's Functions in the Theory of Potential

Published online by Cambridge University Press:  20 January 2009

H. S. Carslaw
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In the Theory of Potential the term Green's Function, used in a slightly different sense by Maxwell, now denotes a function associated with a closed surface S, with the following properties:—

(i) In the interior of S, it satisfies ∇2V = 0.

(ii) At the boundary of S, it vanishes.

(iii) In the interior of S, it is finite and continuous, as also its first and second derivatives, except at the point (x1, y1,z1).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 31 , February 1912 , pp. 71 - 89
Copyright
Copyright © Edinburgh Mathematical Society 1912

References

page 71 note * Cf. Kneser, Die Integralgleichungen und ihre Anwendungen in der Mathematischen Physik (Braunschweig, 1911).

We shall refer to Kneser's book ia the following pages as Kneser, Inttgralgleichungen.

Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Göttinger Nachrichten, Math.Phys. Klasse, 1904.

Stekloff, Théorie générale des fonctions fondamentales, Ann. de la Fac. des Sc. de Toulouse (2), T. 6, p. 351, 1904.

page 72 note * CfKneser, , Integralgleiehungen, p. 127Google Scholar

page 73 note * CfBocher, , An Introduction to the. Study of Integral Equations (Camb. Tracts, No. 10), p. 57.Google Scholar

page 73 note † CfSchmidt, , Zur Theorie der linearen und nichtlinearen Integralgleichungen, Math. Ann. Bd. 63, p. 449, 1907.Google Scholar

page 74 note * Kneser, , Die Theone der Integralgleichungen und die Darstellung willkiirlicher Funktionen in der mathematischen Physik, Math. Ann. Bd. 63, p. 486, 1907. Also Integralgleichungen, §41.Google Scholar

page 74 note † CfKneser, , Integralgleichungen und Darstellung willkürlicher Funktionen von zwei Variabeln, Rend. Circ. Mat. di Palermo, T. XXVII., p. 117, 1909.Google Scholar

page 74 note ‡ Dougall, , The Determination of Green's Functions by means of Cylindrical and Spherical Harmonics, Proc. Edin. Math. Soc, Vol. XVIII., p. 33, 1900.CrossRefGoogle Scholar

page 74 note § The Green's Functions for a Wedge of any Angle and other Problems in the Conduction of Heat, Proc. Lond. Math. Soc. (2), Vol. VIII., p. 365, 191

page 75 note * Cf.Kneser, , Integralgltichungtn, §27.Google Scholar

page 75 note † An usual

page 76 note * CfGray, and Mathews, , Bessel Functions, p. 242.Google Scholar

page 76 note † CfCarslaw, , Fourier's Series, p. 315.Google Scholar

page 77 note * Cf Fourier's Series, p. 315.

page 77 note † In comparing Dougall's results with those given in this paper, it has to be remembered that his Green's Functions are infinite as

page 78 note * CfMacdonald, , The Electrical Distribution on a Conductor bounded by Two Spherical Surfaces cutting at any Angle, Proc. Lond. Math. Soc, (1) Vol. XXVI., p. 159, 1895.Google Scholar

page 82 note * Hobson, , On a Type of Spherical Harmonics of Unrestricted Dtgrtt, Order, and Argument, Phil. Trans. (A), Vol. 187, p. 451, 1896.Google Scholar

page 83 note * Macdonald, Zeroes of the Spherical Harmonic considered as a function of n, Proc. Lond. Math. Soc. (1), Vol. XXXI., p. 273, 1899.Google Scholar

page 84 note * Kneser, , Integralghichungen, §38.Google Scholar

page 85 note * Cf.Macdonald, , Demonstration of Green's Formula for Electric Density near the Vertex of a Right Cone, Trans. Camb. Phil. Soc, Vol. XVIII., p. 293, 1900. The Green's Function for θ′ = 0 is found in that paper.Google Scholar