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Holomorphic solutions about an irregular singular point of an ordinary linear differential equation

Part of: Qualitative theory General theory in ordinary differential equations

Published online by Cambridge University Press:  09 April 2009

C. E. M. Pearce
C. E. M. Pearce
Affiliation:
Department of Applied MathematicsThe University of Adelaide Adelaide, S. A. 5001, Australia
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Abstract

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It is shown that that an ordinary linear differential equation may possess a holomorphic solution in a neighbourhood of an irregular singular point even though the usual linearly independent solutions corresponding to the two roots of the indicial equation both have zero radius of convergence.

MSC classification

Secondary: 34C05: Location of integral curves, singular points, limit cycles 34A25: Analytical theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 39 , Issue 2 , October 1985 , pp. 178 - 186
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Evans, R. L., ‘Asymptotic and convergent factorial series in the solution of linear ordinary differential equations’, Proc. Amer. Math. Soc. 5 (1954), 89–92.Google Scholar
[2]Fabry, E., Sur les intégrales des éguations différentielles linéaires à coefficients rationnels (Thèse, Paris, 1885).Google Scholar
[3]Forsyth, A. R., Theory of differential equations, Volume IV (Cambridge University Press, Cambridge, 1902).Google Scholar
[4]Hartman, P., Ordinary differential equations (2nd edition, Birkhäuser, Boston, Mass., 1982).Google Scholar
[5]Hille, E., Lectures on ordinary differential equations (Addison-Wesley, Reading, Mass., 1969).Google Scholar
[6]Horn, J., ‘Integration linearer Differentialgleichungen durch Laplacesche Integrale und Fakultätreihen’, Jahresbericht der Deutschen Matematiker-Vereinigung 24 (1915), 309–329.Google Scholar
[7]Horn, J., ‘Laplacesche Integrale, Binomialkoefficientenreihen und Gammaquotientenreihen in der Theorie der linearen Differentialgleichungen’, Math. Zeit. 21 (1924), 85–95.CrossRefGoogle Scholar
[8]Ince, E. L., Ordinary differential equations (Longmans, London, 1927).Google Scholar
[9]Jones, W. B. and Thron, W. J., Continued fractions, analytic theory and applications (Encyclopaedia of mathematics and its applications, Vol. 11, Addison-Wesley, Reading, Mass., 1980).Google Scholar
[10]Khovanskii, A. N., The application of continued fractions and their generalizations to problems in approximation theory (Noordhoff, Groningen, 1963).Google Scholar
[11]Trjitzinsky, W. J., ‘Analytic theory of linear differential equations’, Acta Math. 62 (1934), 167–226.Google Scholar
[12]Trjitzinsky, W. J., ‘Laplace integrals and factorial series in the theory of linear differential and linear difference equations’, Trans. Amer. Math. Soc. 37 (1935), 80–146.CrossRefGoogle Scholar
[13]Turrittin, H. L., ‘Convergent solutions of ordinary linear homogeneous differential equations in the neighbourhood of an irregular singular point’, Acta Math. 93 (1955), 27–66.Google Scholar