Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-17T04:43:48.325Z Has data issue: false hasContentIssue false
  • English
  • Français

Transcendental meromorphic solutions of some algebraic differential equations

Part of: Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  09 April 2009

Katsuya Ishizaki  and
Nobushige Toda
Katsuya Ishizaki
Affiliation:
Department of Mathematics Nippon Institute of Technology 4-1 Gakuendai Miyashiro, Minamisaitama Saitama 345-8501, Japanishi@nit.ac.jp
Nobushige Toda
Affiliation:
Center for General Education Aichi Institute of Technology Yakusa, Toyota-shi Aichi-ken 470-0392, Japantoda3-302@coral.ocn.ne.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we treat transcendental meromorphic solutions of some algebraic differential equations. We consider the number of distinct transcendental meromorphic solutions. Algebraic relations between meromorphic solutions and comparisons of the growth of transcendental meromorphic solutions are also discussed.

MSC classification

Secondary: 30D35: Distribution of values, Nevanlinna theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 2 , October 2007 , pp. 157 - 180
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Bank, S. B., Gundersen, G. G. and Laine, I., ‘Meromorphic solutions of the Riccati differential equation’, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), 369–398.CrossRefGoogle Scholar
[2]Bank, S. B. and Kaufman, R. P., ‘On the growth of meromorphic solutions of the differential equation (y′)m = r(z, y)’, Ada Math. 144 (1980), 223–248.Google Scholar
[3]Elaydi, S., An Introduction to Difference Equations (Springer-Verlag, New York, 1996).CrossRefGoogle Scholar
[4]Gol'dberg, A. A., ‘On single-valued solutions of first-order differential equations’, Ukrain. Mat. Zh. 8(1956), 254–261.Google Scholar
[5]Gross, F., ‘On the equation fn + gn = 1’, Bull. Amer. Math. Soc. 72 (1966), 86–88.CrossRefGoogle Scholar
[6]Hayman, W. K., Meromorphic Functions (Oxford at the Clarendon Press, 1964).Google Scholar
[7]He, Y. Z. and I., Laine, ‘The Hayman-Miles theorem and the differential equation (y′)n = r(z, y)’, Analysis (Munich) 10 (1990), 387–396.Google Scholar
[8]Herold, H., Differentialgleichungen im Komplexen (Vandenhoeck & Ruprecht, Gottingen, 1975).Google Scholar
[9]Hille, E., Ordinary Differential Equations in the Complex Domain (Wiley and Sons, New York–London–Sydney–Toronto, 1976).Google Scholar
[10]Laine, I., ‘On the behavior of the solution of some first order differential equations’, Ann. Acad. Sci. Fenn. Ser. A I 497 (1971).Google Scholar
[11]Laine, I., Nevanlinna Theory and Complex Differential Equations (W. Gruyter, Berlin–New York, 1992).Google Scholar
[12]Nevanlinna, R., Le théorème de Picard-Borel et la theorie desfonctions méromorphes (Gauthier-Villard, Paris, 1929).Google Scholar
[13]Steinmetz, N., Eigenschaften eindeutiger Lösungen gewöhnlicher Differentialgleichungen im Komplexen (Ph.D. Thesis, Karlsruhe, 1978).Google Scholar
[14]Steinmetz, N., ‘Zur theorie der binomischen differentialgleichungen’, Math. Ann. 5 (1979), 263–274.Google Scholar
[15]Valiron, G., ‘Sur la dérivée des fonctions algéroïdes’, Bull. Soc. Math. France 59 (1931), 17–39.Google Scholar
[16]von Rieth, J., Untersuchungen gewisser Klassen gewöhnlicher Differentialgleichungen erster und zweiter Ordnung im Komplexen (Ph.D. Thesis, Technischen Hochschule, Aachen, 1986).Google Scholar
[17]Yosida, K., ‘A generalization of Malmquist's theorem’, Japan J. Math. 9 (1933), 253–256.CrossRefGoogle Scholar