On Continuous Parameter Dependence of Roots of Analytic Functions

Abstract

AbstractLet $${\mathcal {A}}_n(\Omega )$$ A n ( Ω ) be the set of analytic functions on a domain $$\Omega $$ Ω of the complex plane which have n roots in $$\Omega $$ Ω , counted with multiplicity. In this note we consider functions in $${\mathcal {A}}_n(\Omega )$$ A n ( Ω ) which depend continuously on a parameter. A simple short proof shows that the set of roots in the Hausdorff metric depends continuously on the parameter. If the parameter space is connected and all roots are known to lie in one of two disjoint open subsets $$\Omega _1$$ Ω 1 , $$\Omega _2$$ Ω 2 of the complex plane, then the number of the roots, counted with multiplicity, in $$\Omega _1$$ Ω 1 and $$\Omega _2$$ Ω 2 , respectively, is independent of the parameter. Each set of roots generates a unique monic polynomial. It is shown that the map which associates with each function in $${\mathcal {A}}_n(\Omega )$$ A n ( Ω ) the corresponding monic polynomial is continuous when $${\mathcal {A}}_n(\Omega )$$ A n ( Ω ) is equipped with the topology of uniform convergence on compact subsets of $$\Omega $$ Ω . Possible applications are indicated.