Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions

Abstract. We study two extremal problems of geometric function theory introduced by A. A. Goldberg in 1973. For one problem we find the exact solution, and for the second one we obtain partial results. In the process we study the lengths of hyperbolic geodesics in the twice punctured plane, prove several results about them and make a conjecture. Goldberg's problems have important applications to control theory. IntroductionGoldberg [16] studied a class of extremal problems for meromorphic functions. Let F 0 be the set of all holomorphic functions f defined in the rings {z : ρ(f ) < |z| < 1}, omitting 0 and 1, and such that the indices of the curve f ({z : |z| = ρ(f )}) with respect to 0 and 1 are non-zero and distinct.Let F 1 ⊂ F 0 be the subclass consisting of functions meromorphic in the unit disk U. Functions in F 1 can be described as meromorphic functions in U with the property that the numbers of preimages of 0, 1 and ∞, counted with multiplicities, are all finite and pairwise distinct.Let F 2 , F 3 , F 4 be the subclasses of F 1 consisting of functions holomorphic in the unit disk, rational functions and polynomials, respectively. For f in any of these classes F j , 1 ≤ j ≤ 4, we define ρ(f ) as ρ(f ) = sup{|z| : f (z) ∈ {0, 1, ∞}}. Goldberg's constants areGoldberg credits the problem of minimizing ρ(f ) to E. A. Gorin. He proved thatand showed that there exist extremal functions for A 0 and A 2 , but extremal functions for A 1 , A 3 or A 4 do not exist. He also proved the estimates A 0 < 0.0091 and 0.0000038 < A 2 < 0.0319.

show abstract

“…The contents of this section is close to the papers [9] and [14], studying conformal maps of circular polygons.…”

Section: Computation Of the Extremal Function Hmentioning

confidence: 84%

Gol’dberg’s constants

Bergweiler

Erëmenko

2013

JAMA

View full text Add to dashboard Cite

show abstract

“…where ∂ = ∂/∂x and the functions in this expression refer to the corresponding multiplication operators. Thus by (6),…”

Section: Formal Powersmentioning

confidence: 90%

“…Clearlyc 0 = 1. Then by (6) we have recursivelyc (4,5) X (5,0) X (5,2) X (5,4) Figure 1: Construction of X (  ) .…”

Section: Formal Powersmentioning

confidence: 99%

“…These include completeness properties of the "formal powers" used to define the coefficients of the power series [21,22]; relationship to transmutation operators, Darboux and other transformations, and Goursat problems [17,25,27,28]; extension to other number systems (quaternions, etc) [8,9,27] and equations of higher order [15]; relaxation of regularity conditions on the coefficients of the differential equation [5,11]. Further, there have appeared numerous applications to problems in physics and engineering [10,16,18,19,29] as well as in complex analysis [6,24].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Sturm–Liouville equations with several spectral parameters

Porter

2015

Bol. Soc. Mat. Mex.

Self Cite

View full text Add to dashboard Cite

Abstract. We give explicit formulas for a pair of linearly independent solutions of (py, thus generalizing to arbitrary d previously known formulas for d = 1. These are power series in the spectral parameters λ 1 , . . . , λ d (real or complex), with coefficients which are functions on the interval of definition of the differential equation. The coefficients are obtained recursively using indefinite integrals involving the coefficients of lower degree. Examples are provided in which these formulas are used to solve numerically some boundary value problems for d = 2, as well as an application to transmission and reflectance in optics.

show abstract

“…Brown [11] presented a two parameter conformal mapping from a disk to a circular-arc quadrilateral, symmetric with respect to the coordinate axes. Brown and Porter [12] developed a conformal mapping from the circular-arc quadrilaterals with four right angles onto a unit disk. Lui et al [13] proposed a representation of general 2D domains with arbitrary topologies using conformal geometry.…”

Section: Introductionmentioning

confidence: 99%

A more robust multiparameter conformal mapping method for geometry generation of any arbitrary ship section

2014

View full text Add to dashboard Cite

The central problem of strip theory is the calculation of potential flow around 2D sections. One particular method of solutions for this problem is conformal mapping of the body section to the unit circle over which solution of potential flow is available. Here, a new multi-parameter conformal mapping method is presented that can map any arbitrary section onto a unit circle with good accuracy. The procedure for finding the corresponding mapping coefficients is iterative. The suggested mapping technique is proved able to map any chine, bulbous, large and fine sections, appropriately. Several examples of mapping the symmetrical as well as the nonsymmetrical sections have been demonstrated. For the symmetrical and nonsymmetrical sections, the results of the current method are compared against other mapping techniques and the currently produced geometries display good agreement with the actual geometries.

show abstract

Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions

Cited by 12 publications

References 16 publications

Gol’dberg’s constants

Gol’dberg’s constants

On Sturm–Liouville equations with several spectral parameters

A more robust multiparameter conformal mapping method for geometry generation of any arbitrary ship section

Contact Info

Product

Resources

About