Halburd, R;
Wang, J;
(2014)
All admissible meromorphic solutions of Hayman's equation.
International Mathematics Research Notices
, 2015
(18)
pp. 8890-8902.
10.1093/imrn/rnu218.
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Abstract
We find all non-rational meromorphic solutions of the equation $ww"-(w')^2=\alpha(z)w+\beta(z)w'+\gamma(z)$, where $\alpha$, $\beta$ and $\gamma$ are rational functions of $z$. In so doing we answer a question of Hayman by showing that all such solutions have finite order. Apart from special choices of the coefficient functions, the general solution is not meromorphic and contains movable branch points. For some choices for the coefficient functions the equation admits a one-parameter family of non-rational meromorphic solutions. Nevanlinna theory is used to show that all such solutions have been found and allows us to avoid issues that can arise from the fact that resonances can occur at arbitrarily high orders. We actually solve the more general problem of finding all meromorphic solutions that are admissible in the sense of Nevanlinna theory, where the coefficients $\alpha$, $\beta$ and $\gamma$ are meromorphic functions.
| Type: | Article |
|---|---|
| Title: | All admissible meromorphic solutions of Hayman's equation |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1093/imrn/rnu218 |
| Publisher version: | http://dx.oi.org/10.1093/imrn/rnu218 |
| Language: | English |
| Additional information: | © The Author(s) 2014. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. |
| Keywords: | math.CV, math.CV, 34M10, 30D35 |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1412352 |
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