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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Keywords:
Algebraic differential equations,
analytic solutions,
representation of solutions

Article copyright:
© Copyright 1971
American Mathematical Society