Painleve's determinateness theorem extended to proper coverings

Abstract

We extend Painleve's determinateness theorem to first-order ordinary differential equations in the complex domain, with known terms allowed be multivalued in the dependent variable. The multivaluedness is supposed to be resolved by proper coverings.

Keywords

• Painleve's determinateness theorem
• complex differential equations
• Cauchy's problem
• analytic continuation
• singularities
• multi-valued function
• Riemann surface