Numerical Conformal Mapping to One-Tooth Gear-Shaped Domains and Applications

Abstract

We study conformal mappings from the unit disk (or a rectangle) to one-tooth gear-shaped planar domains from the point of view of the Schwarzian derivative, with emphasis on numerical considerations. Applications are given to evaluation of a singular integral, mapping to the complement of an annular rectangle, and symmetric multitooth domains.

Appendix: SPPS Method

The sequence $I_n$ of iterated integrals generated by an arbitrary pair of functions $(q_0,q_1)$ is defined recursively by setting $I_0=1$ identically and for $n\ge 1$,

$$\begin{aligned} I_n(z) = \int _0^z I_{n-1}(\zeta )\,q_{n-1}(\zeta )\,\mathrm{d}\zeta , \end{aligned}$$

(34)

where $q_{n+2j}=q_n$ for $j=1,2,\dots$

Proposition 7.1

[14] Let $\psi _0$ and $\psi _1$ be given, and suppose that $y_\infty$ is a non-vanishing solution of

$$\begin{aligned} y_\infty ''+\psi _0\,y_\infty =\lambda _\infty \psi _1\,y_\infty \end{aligned}$$

on the interval [0, 1], where $\lambda _\infty$ is any constant. Choose $q_0=1/y_\infty ^2$, $q_1=\psi _1\,y_\infty ^2$ and define $X^{(n)}$, $\widetilde{X}^{(n)}$ to be the two 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

$$\begin{aligned} y_1= & {} y_\infty \sum _{k=0}^\infty (\lambda -\lambda _\infty )^k \widetilde{X}^{(2k)}, \nonumber \\ y_2= & {} y_\infty \sum _{k=0}^\infty (\lambda -\lambda _\infty )^k X^{(2k+1)} \end{aligned}$$

(35)

are linearly independent solutions of the equation

$$\begin{aligned} y''+\psi _0y=\lambda \psi _1y \end{aligned}$$

(36)

on [0, 1]. Further, the series for $y_1$ and $y_2$ converge uniformly on [0, 1] for every $\lambda$.

It is a straightforward matter to obtain the appropriate linear combination of solutions with desired 1-jets at $z=0$, for example (0, 1) and (1, 0).