A Numerical Model for the Construction of Finite Blaschke Products with Preassigned Distinct Critical Points

TL;DR

This paper introduces a numerical method to construct finite Blaschke products with specified critical points inside the unit disk, enabling precise control over their critical structure for complex analysis applications.

Contribution

A novel numerical model for constructing finite Blaschke products with preassigned critical points, utilizing a sparse nonlinear system and data transformation techniques.

Findings

Method effectively constructs Blaschke products with given critical points.

Demonstrates high accuracy and efficiency through multiple examples.

Applicable to complex analysis and conformal mapping problems.

Abstract

We present a numerical model for determining a finite Blaschke product of degree $n+1$ having $n$ preassigned distinct critical points $z_1,\dots,z_n$ in the complex (open) unit disk $\mathbb{D}$. The Blaschke product is uniquely determined up to postcomposition with conformal automorphisms of $\mathbb{D}$. The proposed method is based on the construction of a sparse nonlinear system where the data dependency is isolated to two vectors and on a certain transformation of the critical points. The efficiency and accuracy of the method is illustrated in several examples.