Complex Variables 24:145-159 (1994)

© 1994 Gordon and Breach Science Publishers S.A.

AMS 1991: 34A20, 30E15

Abstract: We prove a generalization to C of a well known theorem [Maxwell Rosenlicht, Hardy fields, J. Math. Anal. Appl. 93:297-311 (1983)] that in R, the Riccati equation W'+W ²= F ² with F real and positive on the positive real axis, such that lim _{x -> +} F = +, has a family of solutions asymptotic to F and a unique solution asymptotic to -F. The Riccati equation W'+W ²= F ² in the complex domain, where F is a holomorphic function in a partial neighborhood of infinity D and F (z) -> as z -> in D, has uniformly approximate solutions F and -F. Furthermore, we show that there exist solutions W which are uniformly asymptotic to ± F. We establish criteria for the shape of partial neighborhoods of infinity where this occurs. There is a family of solutions uniformly asymptotic to F and a unique solution uniformly asymptotic to -F or vice versa. The situation is reversed in adjacent neighborhoods. We establish a criterion for determining the particular case. Specific examples are provided for the following cases:

F is of polynomial or iterated logarithmic growth and the region is a sector (this is the classical case),
F = e ^x and the region is a horizontal strip,
F = e ^e x and the region is funnel shaped.

@article{dg:asex,author={Gokhman, D.},
title={An asymptotic existence theorem in {\bf C} for the Riccati equation},
journal={Complex Variables},volume={24},pages={145--159},year={1994}}



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