Zeros of meromorphic solutions of second order linear differential equations

Complex Riccati differential equations revisited

“…Then the set of poles on |z| > r 0 consists of finitely many strings of poles. Each string σ accumulates at some Stokes ray (7) s ν : arg z = θ ν = (2ν + 1)π n + 2 and has counting function…”

“…The solutions are meromorphic in the complex plane, and every non-rational solution has order of growth (2) ̺ = lim sup r→∞ log T (r, w) log r = 1 + n/2 mean type, where the non-negative integer n depends on the coefficients a ν only. The aim of this paper is to refine the results of Wittich and others (Bank [1], Gundersen [5], Hellerstein and Rossi [7,8]; see also Laine's book [9], Chapter 5) on equation (1) and the associated linear differential equation (set a 2 w = −u ′ /u)…”

Second Order Differential Equations with Rational Coefficients

“…We also denote by nNR(r,f) the number of nonreal zeros of a meromorphic function / in a closed disk of radius r centered at the origin. In a recent paper [2], the following theorem is proved.…”

“…Examples in [2] show that if 77 is not a polynomial, the transcendentality of fxf2 cannot be removed; while if 77 is a nonconstant polynomial, fxf2 is always transcendental (cf. [1,Theorem 1]).…”

“…[2, p. 604]) (1.4) fx(z) = eizz~\iz-l) and (1.5) f2(z) = cos z -z~ sinz satisfy (1.1) with 77(z) = 1 -2z-2. It is pointed out in [2] that fxf2 is transcendental while fx has a single nonreal zero and f2 has only real zeros. A natural question raised by Hellerstein, Williamson, Kohs and the authors asks whether there are any reasonable hypotheses on /, and f2 which force 77 to be constant.…”