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王升.非齐次线性微分方程亚纯解的零点和极点分布[J].广西科学,1996,3(1):6-8,38. [点击复制]
- Wang Sheng.On the Distribution of Zeros and Poles of Meromorphic Solutions of Non-homogeneous Linear Differential Equations[J].Guangxi Sciences,1996,3(1):6-8,38. [点击复制]
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| 非齐次线性微分方程亚纯解的零点和极点分布 |
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王升
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| (西江大学数学系, 广东肇庆 526061) |
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| 摘要: |
| 以d(g)表示亚纯函数g(z)的增长级;λ(g),λ(1/g)分别表示g(z)的零点、极点序列收敛指数;λ(g),λ(1/g)分别表示g(z)的不同零点、极点序列收敛指数.定理1设Bj(z)(j=0,1,…,k-1;k≥1)和F(z)为亚纯函数且满足d(F)>maxj=0,1,…,k-1{d(Bj)},又设f(z)为f(k)+Bk-1(z)f(k-1)+…+Bo(z)f=F(z)的一个亚纯函数解.则:(a)若d(f)>d(F),则,max{λ(f),λ(1/f)}.(b)若d(F)为有穷非正整数,则max{λ(f),λ(1/f)}≥max{λ(F),λ(1/F)}.定理2设R(z),Q(z)为非零亚纯函数,且满足max{d(Bo),…,d(Bk-1),d(Q),λ(R)}Y).又设f(z)为f(k)+Bk-1(z)f(k-1)+…+Bo(z)f=R(z)+Q(z)≡Y(z)的亚纯函数解,则:(a)若d(Y)<∞,则,max{λ(f),λ(1/f)}≥max{λ(f),λ(1/Y))}.(b)若d(Y)=∞.则max{λ(f),λ(1/f)}=∞. |
| 关键词: 线性微分方程 亚纯解 零点收敛指数 极点收敛指数 |
| DOI： |
| 投稿时间：1995-06-16 |
| 基金项目:国家自然科学基金资助项目(项目编号1947103)。 |
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| On the Distribution of Zeros and Poles of Meromorphic Solutions of Non-homogeneous Linear Differential Equations |
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Wang Sheng
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| (Dept. of Math. West River University, Zhaoqing, Guangdong, 526061) |
| Abstract: |
| Suppose, d(g) is the order of growth of meromorphic function g(z);λ(g),λ(1/g) are zero-exponent and pole-exponent of convergence of g(z), respectively;λ(g),λ(1/g),are different zero-exponent and pole-exponent of convergence of g(z),respectively. Theorem 1 Suppose Bj(z)(j=0,1,…,k-1;k ≥ 1) and F(z) are meromorphic functions, and satisfy d(F)> maxj=0,1,k-1{d(Bj)}, and suppose f(z) is a meromorphic solution of f(k)+Bk-1(z)f(k-1)+…+Bo(z)f=F(z). Then:(a) if d(f) >d(F), then, max {λ(f),λ(1/f)}=d(f). (b) if d(F) is finite non-positive integer, then, max {λ(f),λ(1/f)} ≥ max {λ(F),λ(1/F)}. Theorem 2 Suppose R(z),Q(z) are non-zero meromorphic solutions, and satisfy max {d(Bo),…,d(Bk-1),d(Q), λ(R)} Y), and suppose f(z) is meromorphic solution of f(k)>+Bk-1, (z)f(k-1)+…+Bo(z)f=R(z)+Q(z)=Y(z). Then:(a) if d(Y)<∞, then, max{λ(f),λ(1/f)} ≥ max{λ(Y),λ(1/Y)}. (b) if d(Y)=∞, then, max{λ(f),λ(1/f)}=∞. |
| Key words: linear differential equation meromorphic solution zero-exponent of convergence pole-exponent of convergence |
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