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V.—On the Connexion between Linear Differential Systems and Integral Equations

Published online by Cambridge University Press:  15 September 2014

E. L. Ince
E. L. Ince
Affiliation:
University of Liverpool
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Extract

This paper summarises the results of an attempt to extend the theory upon which the relationship between linear differential equations and integral equations is based. The case in which the nucleus K(x, s) of the integral equation arises as a Green's function is well known; the nucleus is there characterised by its having discontinuous derivates when x = s. The method here dealt with is virtually an extension of Laplace's and analogous methods for solving linear differential equations by definite integrals, and leads to nuclei which are continuous and have continuous derivates for x = s.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 42 , 1923 , pp. 43 - 53
Copyright
Copyright © Royal Society of Edinburgh 1923

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References

page 43 note * An abstract of the theory as it stood in 1910 is given by Bateman, “Report on the Theory of Integral Equations,” British Association, 1910, § 21. References to later work are given in footnotes in the present paper.

page 43 note † For the general theory of the compatibility of linear differential systems, see Bôcher, Leçons sur les méthodes de Sturm (1917), chap. ii.

page 44 note * Forsyth, , Theory of Differential Equations, vol. iv, p. 252.Google Scholar

page 46 note * Kowalewski, , Einführung in die Determinantentheorie, §§ 22, 156.Google Scholar

page 46 note † Whittaker, : (1) Proceedings International Congress, Cambridge, 1912, i, p. 367Google Scholar; Modern Analysis, § 19·21.

(2) Whittaker, Proc. R.S.E., xxxv (1914), pp. 70–77.Google Scholar

(3) Whittaker, Proc. Lond. Math. Soc. (2), xiv (1915), pp. 260–268CrossRefGoogle Scholar; Modern Analysis, § 23·61.

(4) Whittaker, Proc. Edin. Math. Soc., xxxiii (1915), pp. 14–23.Google Scholar

Reference may also be made to A. Milne's integral equation for the parabolic cylinder functions, Proc. Edin. Math. Soc., xxxii (1914), p. 8.

page 47 note * It is virtually the equation exhaustively treated by Abraham, M., Math. Ann., lii (1901), pp. 81–112.Google Scholar

page 47 note † Loc. cit. (1).

page 48 note * Bateman, , Trans. Camb. Phil. Soc., xxi (1909), p. 187.Google Scholar

page 49 note * This development is due to Schmidt, E., Inaugural Dissertation, Göttingen, 1905Google Scholar: see Lalesco, Théorie des équations intégrales, p. 96, or Goursat, , Cours d'analyse, iii, p. 470.Google Scholar

page 51 note * It is not here a question of the development of the nucleus according to its fundamental functions.

page 52 note * Whittaker, loc. cit. (4), p. 22.