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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A representation theorem for large and small analytic solutions of algebraic differential equations in sectors


Author: Steven Bank
Journal: Trans. Amer. Math. Soc. 159 (1971), 293-305
MSC: Primary 34.06
DOI: https://doi.org/10.1090/S0002-9947-1971-0283272-2
MathSciNet review: 0283272
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Abstract: In this paper, we treat first-order algebraic differential equations whose coefficients belong to a certain type of function field. In the particular case where the coefficients are rational functions, our main result states that for any given sector $ S$ in the plane, there exists a positive real number $ N$, depending only on the equation and the angle opening of $ S$, such that any solution $ y(z)$, which is meromorphic in $ S$ and satisfies the condition $ {z^{ - N}}y \to \infty $ as $ z \to \infty $ in $ S$, must be of the form $ \exp \int {c{z^m}(1 + o(1))} $ in subsectors, where $ c$ and $ m$ are constants. (From this, we easily obtain a similar representation for analytic solutions in $ S$, which are not identically zero, and for which $ {z^K}y \to 0$ as $ z \to \infty $ in $ S$, where the positive real number $ K$ again depends only on the equation and the angle opening of $ S$fs


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DOI: https://doi.org/10.1090/S0002-9947-1971-0283272-2
Keywords: Algebraic differential equations, analytic solutions, representation of solutions
Article copyright: © Copyright 1971 American Mathematical Society