J Austral Math Soc Ser A 50 pp258--278, 1991.

Admissable Solutions of the Schwarzian Differential Equation

Katsuya Ishizaki

Abstract

Let R(z, w) be a rational function of w with meromorphic coefficients. It is shown that if the Schwarzian equation

{w, z}m = R(z, w)

(*)

posseses an admissable solution, then d + 2må^l_j=1d(a_j, w) £ 4m, where a_j are distinct complex constants. In particular, when R(z, w) is independent of z, it is shown that if (*) posseses an admissable solution w(z), then by some Möbius transformations u = (aw + b)/(cw + d) (ad - bc ¹ 0), the equation can be reduced to one of the following forms:

{u, z} = C  (u - s1)(u - s2)(u - s3)(u - s4) (u - t1)(u - t2)(u - t3)(u - t4) ,

{u, z}3 = C  [(u - s1)3(u - s2)3] [(u - t1)3(u - t2)2(u - t3)] ,

{u, z}3 = C  [(u - s1)3(u - s2)3] [(u - t1)2(u - t2)2(u - t3)2] ,

{u, z}2 = C  [(u - s1)2(u - s2)2] [(u - t1)2(u - t2)(u - t3)] ,

{u, z} = C  [(u - s1)(u - s2)] [(u - t1)(u - t2)] ,

{u, z} = C,

where t_j (j = 1, ..., 4) are distinct constants, and s_j (j = 1, ..., 4) are constants, not necessarily distinct.

1980 AMS Subject Classification (1985 Revision): primary 34C10; secondary 30D35

Browse the article

Authors

Katsuya Ishizaki

Department of Mathematics, Tokyo National College of Technology, 1220 -2 Kunugida-cho, Hachioji, Tokyo 193, Japan.