| 具有全纯系数的二阶线性微分方程解的零点分布 |
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| 引用本文： | 蔡惠京.具有全纯系数的二阶线性微分方程解的零点分布[J].广东广播电视大学学报,2008,17(4):97-100. |
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| 作者姓名： | 蔡惠京 |
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| 作者单位： | 中山广播电视大学,广东中山,528402 |
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| 摘 要： | 本文研究具有超越整函数系数的二阶线性微分方程f″＋A（z）f=^0的解的零点分布。证明当A（%）的增长级为（2，1．p）时，方程的每一个非平凡解的增长级都为（3，1．p），而且总存在一个非平凡解f（z）的零点收敛级等于其增长级（3，1;p）。进一步给出了方程存在无零点解的条件，证明当P非为整数时，方程的两个线性无关解中至多只有一个无零点。最后，证明了该方程总存在两个线性无关解f1（z）和f2（z），使得f1（z）×f2（z）的零点收敛级等于其增长级（3，1；P）。
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| 关 键 词： | 微分方程 整函数 增长级 零点分布 |
Distribution of Zeros of the Solutions of Second Order Linear Differential Equations with Entire Coefficients |
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| CAI Hui-jing.Distribution of Zeros of the Solutions of Second Order Linear Differential Equations with Entire Coefficients[J].Journal of Guangdong Radio & Television University,2008,17(4):97-100. |
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| Authors: | CAI Hui-jing |
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| Institution: | CAI Hui-jing(Zhongshan Radio & TV University,Zhongshan,Guangdong,China,528402) |
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| Abstract: | In this paper, we investigate the distribution of zeros of the solutions of second order linear differential equations f″ ＋ A （ z ） f = 0 with transcendental coefficients. It is proved that if A （z） has order （ 2,1 ;p）, then the order of growth of nontrivi- al solution f is （3,1 ;p） and the equation always has a solution f that the exponent of convergence of its zero- sequence is （3, 1 ;p） too. Further, we give conditions that the equation has some solutions which have no zeros. It is proved that if p is not an in- teger, then at most one of the solutionsfl （z） ,f2 （z）, which are two linearly independent solutions of the equation, has no zeros. Finally, it is proved that the equation always has two linearly independent solutions fl （z） and f2 （z） which has the property that the exponent of convergence of zero - sequence of fl （z） × f2 （z） is （ 3,1 ;p）. |
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| Keywords: | differential equations entire function order of growth distribution of zeros |
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