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

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