2014/11/06 by Rod Halburd, Jun Wang, Halburd, Rod +1
Mathematics · #30D35 #34M10 #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions #math.CV #msc:30D35 #msc:34M10
paper · pdf · doi:10.48550/arxiv.1411.1796
11 pages
arxiv created 2014/11/06 · openalex publication_date 2014/11/06 · arxiv updated 2014/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We find all non-rational meromorphic solutions of the equation ww"-(w')2=α(z)w+β(z)w'+γ(z), where α, β and γ 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 α, β and γ are meromorphic functions.