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Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions

Philip R. Brown, R. Michael Porter

Abstract

Numerical and theoretical aspects of conformal mappings from a disk to a circular-arc quadrilateral, symmetric with respect to the coordinate axes, are developed. The problem of relating the accessory parameters (prevertices together with coefficients in the Schwarzian derivative) to the geometric parameters is solved numerically, including the determination of the parameters for univalence. The study involves the related mapping from an appropriate Euclidean rectangle to the circular-arc quadrilateral. Its Schwarzian derivative involves the Weierstrass P-function, and consideration of this related mapping problem leads to some new formulas concerning the zeroes and the images of the half-periods of P.

Conformal Mapping of Circular Quadrilaterals and Weierstrass Elliptic Functions

Abstract

Numerical and theoretical aspects of conformal mappings from a disk to a circular-arc quadrilateral, symmetric with respect to the coordinate axes, are developed. The problem of relating the accessory parameters (prevertices together with coefficients in the Schwarzian derivative) to the geometric parameters is solved numerically, including the determination of the parameters for univalence. The study involves the related mapping from an appropriate Euclidean rectangle to the circular-arc quadrilateral. Its Schwarzian derivative involves the Weierstrass P-function, and consideration of this related mapping problem leads to some new formulas concerning the zeroes and the images of the half-periods of P.

Paper Structure

This paper contains 14 sections, 5 theorems, 69 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Description of problem
  3. Parameters of s.c.q. s
  4. Schwarzian derivative and accessory parameters
  5. Mapping from a rectangle
  6. The Schwarzian derivative and the $\wp$-function
  7. The zeros of $\wp$ expressed as elliptic integrals
  8. Calculation of the image quadrilateral as a function of the accessory parameter
  9. SPPS method
  10. $p_2$ and $\kappa_1$
  11. Parameters for univalence
  12. Interval of univalence for fixed $\alpha$,$t$
  13. Estimates for $\lambda^*_{\rm min}$, $\lambda^*_{\rm max}$.
  14. Universal cover of a doubly-punctured disk

Key Result

Proposition 3.1

KP1 Let $\psi_0$ and $\psi_1$ be given, and suppose that $y_\infty$ is a nonvanishing solution of on the interval $[0,1]$. Choose $q_0=1/y_\infty^2$, $q_1=\psi_1\,y_\infty^2$ and define $X^{(n)}$, $\widetilde{X}^{(n)}$ to be the sequences of iterated integrals generated by $(q_0,q_1)$ and by $(q_1,q_0)$ respectively. Then for each $\lambda\in\mathbb{C}$ the functions are linearly independent sol

Figures (8)

  • Figure 1: Geometric parameters defining symmetric circular quadrilaterals.
  • Figure 2: Extremal domains for $\alpha=0.2$, $t=\pi/6$, with $\lambda_{\rm min}=-0.479608$ (left) and $\lambda_{\rm max}=1.30611$ (right). Values near the vertices were plotted by extrapolation.
  • Figure 3: Extremal domains for $\alpha=1.3$, $t=3\pi/8$, with $\lambda_{\rm min}=-1.28089$ (left) and $\lambda_{\rm max}= 1.84854$ (right).
  • Figure 4: Comparison of upper and lower bounds obtained for $\mu_{\rm min}$, $\mu_{\rm max}$ for $\alpha=0.25$ and $0.1\le\tau/i\le2.0$, given by Proposition \ref{['prop:mubounds']} (solid lines) and Proposition \ref{['prop:nehari']} (dashed lines).
  • Figure 5: Values of $\hbox{arccot}\,4\lambda$ for which the mapping $f_{\alpha,t,\lambda}$ ($0<t<\pi/2$) is univalent in the unit disk. Bounds derived from $\mu^*_{\rm min}$,$\mu^*_{\rm max}$ are shown with a solid gray line; bounds derived from Nehari's theorem with a dashed line.
  • ...and 3 more figures

Theorems & Definitions (5)

  • Proposition 3.1
  • Lemma 4.1
  • Proposition 4.2
  • Corollary 4.3
  • Proposition 4.4