An unsolvable Cousin problem
HTML articles powered by AMS MathViewer
- by Lee A. Rubel PDF
- Proc. Amer. Math. Soc. 104 (1988), 410-412 Request permission
Abstract:
We construct a Laurent series $\sum \nolimits _{n = - \infty }^\infty {{A_n}{z^n}}$, convergent in an annulus $\left \{ {\rho < \left | z \right | < 1/\rho } \right \}$ for some $\rho < 1$, which satisfies an algebraic differential equation (ADE) there, but such that neither of the series $\sum \nolimits _{n = 0}^\infty {{A_n}{z^n}}$ and $\sum \nolimits _{n = - \infty }^{ - 1} {{A_n}{z^n}}$ satisfies any ADE.References
- Steven B. Bank, On the hypertranscendence of meromorphic solutions of certain functional equations, Proc. Roy. Soc. Edinburgh Sect. A 83 (1979), no. 1-2, 45–54. MR 538584, DOI 10.1017/S0308210500011367
- Michael Boshernitzan and Lee A. Rubel, Coherent families of polynomials, Analysis 6 (1986), no. 4, 339–389. MR 877792, DOI 10.1524/anly.1986.6.4.339
- Leonard Lipshitz and Lee A. Rubel, A differentially algebraic replacement theorem, and analog computability, Proc. Amer. Math. Soc. 99 (1987), no. 2, 367–372. MR 870803, DOI 10.1090/S0002-9939-1987-0870803-1
- Alexander Ostrowski, Über Dirichletsche Reihen und algebraische Differentialgleichungen, Math. Z. 8 (1920), no. 3-4, 241–298 (German). MR 1544442, DOI 10.1007/BF01206530 G. Pólya, Über das Anwachsen von ganzen Funktionen die einer Differentialgleichung genügen, Vierteljahrschrift Naturforsch. Gesellschaft Zürich 61 (1916), 531-545. —, Un pendant du théorème d’approximation de Liouville dans la théorie des équations différentielles, Enseign. Math. 19 (1917), 96-97. Jan Popken, Über arithmetische Eigenschaften analytischer Funktionen, North-Holland, Amsterdam, 1935.
- Lee A. Rubel, Some mathematical limitations of the general-purpose analog computer, Adv. in Appl. Math. 9 (1988), no. 1, 22–34. MR 930770, DOI 10.1016/0196-8858(88)90004-8
- Stanisław Saks and Antoni Zygmund, Analytic functions, Monografie Matematyczne, Tom XXVIII, Polskie Towarzystwo Matematyczne, Warszawa-Wroclaw, 1952. Translated by E. J. Scott. MR 0055432
- Georges Valiron, Fonctions analytiques, Presses Universitaires de France, Paris, 1954 (French). MR 0061658
- Hans Wittich, Bemerkung zu einer Funktionalgleichung von H. Poincaré, Arch. Math. (Basel) 2 (1950), 90–95 (German). MR 33353, DOI 10.1007/BF02038566
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 410-412
- MSC: Primary 34A20; Secondary 30D05
- DOI: https://doi.org/10.1090/S0002-9939-1988-0962806-4
- MathSciNet review: 962806