Kecker, T; (2014) On the singularity structure of differential equations in the complex plane. Doctoral thesis , UCL (University College London).

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Abstract

In this dissertation the structure of singularities in the complex plane of solutions of certain classes of ordinary differential equations and systems of equations is studied. The thesis treats two different aspects of this topic. Firstly, we introduce the concept of movable singularities for first and second-order ordinary differential equations. On the one hand the local behaviour of solutions about their movable singularities is investigated. It is shown, for the classes of equations considered, that all movable singularities of all solutions are either poles or algebraic branch points. That means locally, about any movable singularity z0, the solutions are finitely branched and represented by a convergent Laurent series expansion in a fractional power of z-z0 with nite principle part. This is a generalisation of the Painleve property under which all solutions have to be single-valued about all their movable singularities. The second aspect treated in the thesis deals with the global structure of the solutions. In general, the solutions of the equations discussed in the first part have a complicated global behaviour as they will have infinitely many branches. In the second part conditions are discussed for certain equations under the existence of solutions that are globally nite-branched, leading to the notion of algebroid solutions. In order to do so, some concepts from Nevanlinna theory, the value-distribution theory of meromorphic functions and its extension to algebroid functions are introduced. Then, firstly, Malmquist's theorem for first-order rational equations with algebroid solutions is reviewed. Secondly, certain second-order equations are considered and it is examined to what types of equations they can be reduced under the existence of an admissible algebroid solution.

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