# On the Zero Distribution of Solutions of Second Order Linear Differential Equations in the Complex Domain

### New Zealand Journal of Mathematics

Vol. 42, (2012), Pages 9-16

Jianren Long

School of Mathematics and Computer Science,

Guizhou Normal University,

Guiyang, 550001, P. R. China

Pengcheng Wu

School of Mathematics and Computer Science,

Guizhou Normal University,

Guiyang, 550001, P. R. China

Xiubi Wu

School of Mathematics and Computer Science,

Guizhou Normal University,

Guiyang, 550001, P. R. China

Abstract Let $f(z)\not \equiv 0$ be a solution of f'' + P(z)f = 0, where P(z) is a polynomial. Then the set of accumulation lines of zero-sequence is a subset of the Borel directions of f(z). Let f1 and f2 be two linearly independent solutions of f'' + P(z)f = 0, where P(z) is a polynomial of degree n and set E = f1f2. Then, for every accumulation line $\arg z =\theta$ of zero-sequence of E, there is another accumulation line $\arg z =\phi$ of zero-sequence of E such that $|\phi-\theta|=\frac{2\pi}{n+2}$.

Keywords Differential equation, accumulation line of zero-sequence, exponent of convergence of zero-sequence, angular distribution.

Classification (MSC2000) 34M10; 30D35.