MATHEMATICS

B. A. VERTGEIM

ON THE APPROXIMATE CONSTRUCTION OF CERTAIN CONFORMAL MAPPINGS

(Presented by Academician M. A. Lavrent′ev, 15 X 1957)

It is known that the problem of conformally mapping the circle \(|z|<1\) onto a domain of the \(w\)-plane containing the point \(w_0=0\) and bounded by a simple, closed contour \(L\), star-shaped with respect to the point \(w_0\), given by the equation

\[ w=\exp\{f(t)+it\}, \qquad f(t+2\pi)=f(t), \tag{1} \]

under the condition that

\[ w(0)=0, \qquad w'(0)>0, \tag{2} \]

reduces to a nonlinear singular integral equation for the function \(t=t(\varphi)\), \(t(\varphi+2\pi)=t(\varphi)+2\pi\):

\[ t(\varphi)-\varphi=Sf[t(\varphi)]; \tag{3} \]

here and in what follows \(S\) denotes the integral operator with Hilbert kernel

\[ Sh(\varphi)=\frac{1}{2\pi}\int_0^{2\pi} h(\sigma)\operatorname{ctg}\frac{\varphi-\sigma}{2}\,d\sigma. \tag{4} \]

Equation (3) admits the application of iterations

\[ t_{n+1}(\varphi)=\varphi+Sf[t_n(\varphi)] \]

(Theodorsen’s method). The convergence of the process was proved in the work \((^1)\) under the additional assumption that

\[ |f'(t)|\leq q<1. \tag{5} \]

In the present note we consider the case of a non-star-shaped contour \(L\), defined by the equation

\[ w=\exp\{f_1(t)+if_2(t)\}, \tag{6} \]

where

\[ f_1(t+2\pi)=f_1(t), \qquad f_2(t+2\pi)=f_2(t)+2\pi, \]

\[ (f_1')^2+(f_2')^2\geq d^2>0. \]

It is assumed that \(f_1\) and \(f_2\) are twice continuously differentiable functions, and moreover their second derivatives satisfy the Lipschitz condition

\[ |f_k''(t_1)-f_k''(t_2)|\leq l_k|t_1-t_2|, \qquad k=1,2. \tag{7} \]

The problem analogous to the preceding one, with condition (2), can be reduced to the nonlinear singular integral equation

\[ P[t(\varphi)]\equiv f_2[t(\varphi)]-\varphi-Sf_1[t(\varphi)]=0. \tag{8} \]

Theodorsen’s method is inapplicable here, since for an unknown contour the function \(f_2\) is not monotone. It turns out to be possible to apply, for solving this equation (and also in the preceding case—without restriction (5)), Newton’s method, developed by L. V. Kantorovich for the approximate solution of nonlinear functional equations \((^{2,3})\). With this method the successive approximations are determined from linear singular integral equations (we give them for the modified method)

\[ f'_2[t_0](t_{n+1}-t_n)-S f'_1[t_0](t_{n+1}-t_n)=-P[t_n]. \tag{9} \]

Here \(n=0,1,2,\ldots\). Solving this equation explicitly, we arrive at algorithm (14) (see below). To solve equation (9), whose index \((^4)\) turns out to be equal to zero, we transform it into a Hilbert problem, somewhat modifying the technique for solving the equation conjugate with the characteristic one ((\(^4\)), p. 133).

Let \(H_\mu\) be the space of \(2\pi\)-periodic functions \(u(\varphi)\) satisfying the Hölder condition, and let the norm in it be defined in the usual way by \(\|u\|=\max |u|+A_u\), where \(A_u\) is the Hölder constant. Introduce the notation:

\[ f'_2(t)-i f'_1(t)=g(t)e^{i\omega(t)}, \qquad g(t)\ge d>0, \]

\[ \frac{\cos\omega[t_0(\varphi)]}{g[t_0(\varphi)]}=h_1(\varphi), \qquad \frac{\exp\{-S\omega[t_0(\varphi)]\}}{g[t_0(\varphi)]}=h_2(\varphi), \]

\[ \exp\{+S\omega[t_0(\varphi)]\}\sin\omega[t_0(\varphi)]=h_3(\varphi), \]

\[ \|P[t_0(\varphi)]\|=\bar{\eta}_0; \qquad \frac{1}{2\pi}\int_0^{2\pi}\omega[t_0(\varphi)]\,d\varphi=\omega_c, \tag{10} \]

\[ B_0=\|h_1(\varphi)\|+\|h_2(\varphi)\|\|h_3(\varphi)\|\bigl(N_0+\operatorname{tg}|\omega_c|\bigr), \]

\[ K=(l_1+N_0 l_2)(2\pi+\|t_0(\varphi)-\varphi\|+2B_0\bar{\eta}_0). \]

Here \(N_0\) is an estimate for the norm of the linear operator \(S\) (see (4)), which, by a known theorem of Privalov, maps \(H_\mu\) into itself; for example, one can show that

\[ \|S\|\le N_0=\max\left[ \frac{1}{\mu\pi}(1+2^{1+\mu}+3^\mu)+\frac{\pi}{1-\mu}; \frac{4^{1+\mu}}{\mu\pi} \right]+\frac{2\pi^\mu-1}{\mu}. \]

Applying theorem 1 of the paper \((^5)\) (for \(\alpha=1\)), we arrive at the following result.

Theorem. Suppose that, for equation (8) and the initial approximation \(t_0(\varphi)\), the conditions

\[ \cos\omega_c\ne 0, \tag{11} \]

\[ h_0=B_0^2K\bar{\eta}_0<\tfrac{1}{2}. \tag{12} \]

are satisfied. Then in the domain

\[ \|t(\varphi)-t_0(\varphi)\|\le 2B_0\bar{\eta}_0 \tag{13} \]

there exists a solution \(t^*(\varphi)\) of equation (8), and the successive approximations \(\{t_n(\varphi)\}\):

\[ t_{n+1}(\varphi)=t_n(\varphi)-h_1(\varphi)P[t_n(\varphi)]+ \]

\[ +h_2(\varphi)\left\{ S\bigl(h_3(\varphi)P[t_n(\varphi)]\bigr) -\frac{\operatorname{tg}|\omega_c|}{2\pi} \int_0^{2\pi}h_3(\varphi)P[t_n(\varphi)]\,d\varphi \right\} \tag{14} \]

converge to \(t^*(\varphi)\):

\[ \left\|t_n(\varphi)-t^*(\varphi)\right\|\leq \frac{q^n}{1-q} B_0\overline{\eta}_0,\qquad q=1-\sqrt{1-2h_0}<1. \tag{15} \]

Remark. Condition (11) is always satisfied if the contour \(L\) is star-shaped or if it has an axis of symmetry (in the latter case \(\omega_c=0\)). In the general case one can prove that condition (11) will be satisfied if

\[ |t_0(\varphi)-t_i^*(\varphi)|\leq \frac{\pi}{2\max |\omega'(t)|}. \]

Condition (12) is also satisfied in some neighborhood of the solution \(t^*(\varphi)\), whose existence in the space under consideration is evident. We also note that the functions \(t(\varphi)\in H_\mu\); however, the substitution \(t(\varphi)-\varphi=u(\varphi)\) leads to elements \(u(\varphi)\in H_\mu\). The functions \(t(\varphi)\) have been retained in the formulas for the sake of brevity; finally, \(t(\varphi)-t_0(\varphi)\in H_\mu\).

Other results on Newton’s method are applied in a similar way, for example, the convergence theorems for the basic method.

The author expresses his deep gratitude to Prof. L. I. Volkovyskii for scientific guidance.

Perm Mining Institute

Received
6 IX 1957

REFERENCES

¹ S. Warschawski, Quart. Appl. Math., 3, No. 1, 12 (1945).
² L. V. Kantorovich, Uspekhi Mat. Nauk, 3, issue 6, 89 (1948).
³ L. V. Kantorovich, DAN, 80, No. 6, 849 (1951).
⁴ N. I. Muskhelishvili, Singular Integral Equations, 1946.
⁵ B. A. Vertgeim, DAN, 110, No. 5, 719 (1956).