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

@article{Glader2018ANM, title={A Numerical Model for the Construction of Finite Blaschke Products with Preassigned Distinct Critical Points}, author={Christer Glader and Ray Porn}, journal={arXiv: Numerical Analysis}, year={2018} }

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…

## References

SHOWING 1-10 OF 16 REFERENCES

### Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problems

- Mathematics
- 2017

The determination of a finite Blaschke product from its critical points is a well-known problem with interrelations to several other topics. Though existence and uniqueness of solutions are…

### Critical Points, the Gauss Curvature Equation and Blaschke Products

- Mathematics
- 2013

In this survey paper we discuss the problem of characterizing the critical sets of bounded analytic functions in the unit disk of the complex plane. This problem is closely related to the…

### Rational Functions with Prescribed Critical Points

- Mathematics
- 2002

Abstract.A rational function is the ratio of two complex polynomials in one variable without common roots. Its degree is the maximum of the degrees of the numerator and the denominator. Rational…

### Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem

- Mathematics
- 2008

We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation (PDE) Δ u=4 e2u and combine this result with methods from nonlinear…

### On a Class of Conformal Metrics

- MathematicsNagoya Mathematical Journal
- 1962

Last year when I was preparing for course lectures the work of Ahlfors [1] which establishes that the Bloch constant is at least as large as it appeared to me that the resources of the theory of…

### Note on the location of zeros of the derivative of a rational function whose zeros and poles are symmetric in a circle

- Mathematics
- 1939

radius 1/n having restrictions at hv. Let F denote the family of all collections G(„,n). It can easily be shown that -Fis the required family. Axiom 2 is evidently satisfied. Note. If in space T the…

### Introduction to Circle Packing: The Theory of Discrete Analytic Functions

- Mathematics
- 2005

Part I. An Overview of Circle Packing: 1. A circle packing menagerie 2. Circle packings in the wild Part II. Rigidity: Maximal Packings: 3. Preliminaries: topology, combinatorics, and geometry 4.…

### On Critical Points of Proper Holomorphic Maps on The Unit Disk

- Mathematics
- 1996

We prove that a proper holomorphic map on the unit disk in the complex plane is uniquely determined up to post‐composition with a Möbius transformation by its critical points. 1991 Mathematics…

### Introduction to circle packing: the theory of discrete analytic functions

- Mathematics
- 2007

INTRODUCTION TO CIRCLE PACKING THE THEORY OF DISCRETE ANALYTIC FUNCTIONS PDF Are you looking for Ebook introduction to circle packing the theory of discrete analytic functions PDF? You will be glad…

### Critical Points

- HistoryNature
- 1972

Introduction to Phase Transitions and Critical Phenomena.By H. E. Stanley. Pp. xx + 308. (Clarendon: Oxford; Oxford University: London, July 1971.) £5.