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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Kodai Mathematical Journal
( Vol. 15 NO.2 / (1992))
On the Frequency of Complex Zeros of Solutions of Certain Differential Equations
Steven B. Bank,
Pages. 165-184
Abstract In this paper, we investigate the frequency of zeros of solutions of linear differential equations of the form
$w^{(k)}+sumlimits_{j=1}^{k-1}Q_jw^{(j)}+(Q_0+Re^P)w=0,$ where $k
geqq 2$, and where $Q_0,cdots, Q_{k-1},R$ and $P$ are arbitrary
polynomials with $R
otequiv 0$ and $P$ non-constant. All
solutions $f
otequiv 0$ of such an equation are entire functions
of infinite order of growth, but there are examples of such
equations which can possess a solution whose zero-sequence has a
finite exponent of convergence. In this paper, we show that unless
a special relation exists between the polynomials
$Q_0,cdots,Q_{k-1}$, and $P$, all solutions of such an equation
have an infinite exponent of convergence for their zero-sequences.
This result extends earlier results for the equation,
$w^{(k)}+(Q_0+Re^P)w=0.$
Contents 1. Introduction
2. Preliminaries for main result
3. Main result
4. Remarks and examples
5. Concepts from the strodt theory [17]
6. A result from [2]
7. Main lemma on asymptotic integration
8. Proof of the main result
9. Examples
Key words
Mathmatical Subject Classification