Geometry of transcendental singularities of complex analytic functions and vector fields

Abstract

On Riemann surfaces MM, there exists a canonical correspondence between a possibly multivalued function ΨX{\Psi }_{X} whose differential is single-valued (i.e. an additively automorphic singular complex analytic function) and a vector field XX. From the point of view of vector fields, the singularities that we consider are zeros, poles, isolated essential singularities, and accumulation points of the above. The theory of singularities of the inverse function ΨX‒1{\Psi }_{X}^{‒1} is extended from meromorphic functions to additively automorphic singular complex analytic functions. The main contribution is a complete characterization of when a singularity of ΨX−1{\Psi }_{X}^{-1} is algebraic, is logarithmic, or arises from a zero with non-zero residue of XX. Relationships between analytical properties of ΨX{\Psi }_{X}, singularities of ΨX−1{\Psi }_{X}^{-1} and singularities of XX are presented. Families and sporadic examples showing the geometrical richness of vector fields on the neighbourhoods of the singularities of ΨX−1{\Psi }_{X}^{-1} are studied. As applications, we have; a description of the maximal univalence regions for complex trajectory solutions of a vector field XX, a geometric characterization of the incomplete real trajectories of a vector field XX, and a description of the singularities of the vector field associated with the Riemann ξ{\rm{\xi }}-function.

Keywords

• complex analytic vector fields
• riemann surfaces
• essential singularities
• transcendental singularities
• 32s65
• 58k45
• 30f25
• 30d30
• 30d20