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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Finite order solutions of second order linear differential equations


Author: Gary G. Gundersen
Journal: Trans. Amer. Math. Soc. 305 (1988), 415-429
MSC: Primary 34A20; Secondary 30D15, 34A30, 34C11
DOI: https://doi.org/10.1090/S0002-9947-1988-0920167-5
MathSciNet review: 920167
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Abstract: We consider the differential equation $ f'' + A(z)f' + B(z)f = 0$ where $ A(z)$ and $ B(z)$ are entire functions. We will find conditions on $ A(z)$ and $ B(z)$ which will guarantee that every solution $ f\not \equiv 0$ of the equation will have infinite order. We will also find conditions on $ A(z)$ and $ B(z)$ which will guarantee that any finite order solution $ f\not \equiv 0$ of the equation will not have zero as a Borel exceptional value. We will also show that if $ A(z)$ and $ B(z)$ satisfy certain growth conditions, then any finite order solution of the equation will satisfy certain other growth conditions. Related results are also proven. Several examples are given to complement the theory.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0920167-5
Keywords: Linear differential equation, entire function, finite order of growth, exponent of convergence of the zero-sequence
Article copyright: © Copyright 1988 American Mathematical Society