UDC 517.54

MATHEMATICS

B. G. BAIBARIN

ON THE QUESTION OF CONFORMAL MAPPING OF CIRCULAR DOMAINS

(Presented by Academician M. A. Lavrent’ev on 28 III 1966)

1. Let \(L\) be a simple nonclosed curve composed of a finite number of arcs of circles and line segments, beginning at a point \(\zeta_0>0\) of the real axis, not passing through the origin, and not intersecting the ray \(l:\ \zeta_0<z<\infty\).

Notation.

1)
\[ z=\Phi(w,t),\quad \Phi(\beta(t),t)=0,\quad \Phi(\infty,t)=\infty, \tag{1} \]
is a function conformally mapping the half-plane \(\operatorname{Im} w>0\) onto the domain \(B(t)\), obtained from the plane \((z)\) by making a slit along the curve \(\mathcal L(t)+l\), where \(\mathcal L(t): z=\xi(\tau),\ t_0\le \tau\le t,\ t<T\), is a part of the curve \(L\).

2) \(a_p(t),\ p=0,1,2,\ldots,n,\) are the preimages of the corner points \(z_i,\ i=0,1,\ldots,\ldots,m;\ n=2m;\ z_0=\zeta_0\) of the curve \(\mathcal L(t)+l\), including also the end of the slit, whose preimage we consider to be \(a_0(t)\) (to each non-end corner point \(z_i\) there correspond two points among the \(a_p(t)\)) under the mapping (1).

3) \(\alpha_p\pi,\ p=0,1,2,\ldots,n,\) is the angle between the tangents on the left and on the right to the curve \(\mathcal L(t)+l\) at the points \(z_i\).

4) \(M_p(t),\ p=0,1,2,\ldots,n,\) are the accessory constants of the Schwarz derivative \(S(w,t)\) for the function \(\Phi(w,t)\) (3).

5) \(t_i,\ i=0,1,2,\ldots,m,\) are the values of the parameter \(t\) corresponding to the corner points \(z_i,\ z_i=\xi(t_i)\).

Theorem. Let \(\chi(w)\) be a function conformally mapping the half-plane \(\operatorname{Im} w>0\) onto the plane with a slit along the ray \(l\). Then the functions
\[ \Psi(w,t)=\log \Phi_w'(w,t),\qquad S(w,t)=\Psi''(w,t)-\frac12\Psi'^2(w,t) \]
are integrals of the equations
\[ \frac{\partial\Psi(w,t)}{\partial t} +\frac{1}{w-a_0(t)}\frac{\partial\Psi(w,t)}{\partial w} = \frac{1}{(w-a_0(t))^2}, \tag{2} \]
\[ \frac{\partial S(w,t)}{\partial t} +\frac{1}{w-a_0(t)}\frac{\partial S(w,t)}{\partial w} -\frac{2S(w,t)}{(w-a_0(t))^2} -\frac{6}{(w-a_0(t))^4} =0, \tag{3} \]
satisfying the initial conditions
\[ \Psi(w,t_0)=\log\chi'(w),\qquad S(w,t_0)=\Psi''(w,t_0)-\frac12\Psi'^2(w,t_0). \]

1. Denote by \(\omega(t,t_m)\) the integral of the equation
   \[ \frac{d\omega(t,t_m)}{dt} = \frac{1}{\omega(t,t_m)-a_0(t)}, \tag{4} \]
   which for \(t=t_m\) takes the value \(a_0(t_m)\). Then equation (4) has:

a) a unique integral \(\omega_1(t,t_m)\), \(\omega_1(t_m,t_m)=a_0(t_m)\), for which
\[ \lim_{t\to t_m+0} \frac{\omega_1(t,t_m)-a_0(t)}{\sqrt{t-t_m}} = \sqrt{\frac{2\alpha_1}{\alpha_n}}; \]

b) a unique integral \(\omega_2(t,t_m)\), \(\omega_2(t_m,t_m)=a_0(t_m)\), for which
\[ \lim_{t\to t_m+0} \frac{\omega_2(t,t_m)-a_0(t)}{\sqrt{t-t_m}} = -\sqrt{\frac{2\alpha_n}{\alpha_1}}. \]

Moreover, the relation

\[ \lim_{t\to t_m+0}\frac{\omega_1(t,t_m)-a_0(t)}{\omega_2(t,t_m)-a_0(t)} =-\frac{\alpha_1}{\alpha_n} \]

is valid.

3. The curvature \(\varkappa(t)\) of the curve \(\mathcal L(t)+l\) at the point \(z=\zeta(t)\) is determined by the formula

\[ s'(t)\varkappa(t)=\operatorname{Im}\left\{ i\,\frac{a_0(t)-\alpha(t)}{\gamma(t)} +\left(\frac{d\bar\beta(t)}{dt}\right)^2 -\frac{d\bar\beta(t)}{dt}\frac{da_0(t)}{dt} + \frac{1}{\pi}\int_{t_0}^{t} \frac{\partial H^*(a_0(t),t,\tau)}{\partial t}\,d\theta(\tau) \right\}, \tag{5} \]

where \(s(t)\) is the length of the curve \(\mathcal L(t)\), measured from the point \(\zeta_0\); \(\theta(t)\) is the angle formed by the tangent to the curve \(\mathcal L(t)\) at the point \(z=\zeta(t)\) with the real axis; \(a(t)=\operatorname{Re}\beta(t)\), \(\gamma(t)=\operatorname{Im}\beta(t)\); \(\beta(t)\) is the integral of equation (4), assuming at \(t=t_0\) the value \(\beta(t_0)=\beta_0\), \(\operatorname{Im}\beta_0>0\), and

\[ H^*(a_0(t),t,\tau) =\log \frac{a_0(t)-\omega_2(t,\tau)}{a_0(t)-\omega_1(t,\tau)} \frac{\omega_1(t,\tau)-\beta(t)}{\omega_2(t,\tau)-\beta(t)} . \]

4. What has been said above, as well as what was set forth in (1), makes it possible to propose a method for determining the parameters of the Schwarz derivative \(S(w)\) for the function \(z=\Phi(w)\), which conformally maps the half-plane \(\operatorname{Im} w>0\) onto a simply connected bounded domain \(B\), whose boundary \(\Gamma\) consists of a finite number of arcs of circles and line segments. It may be assumed, without loss of generality, that the domain \(B\) contains the origin and that one of the corner points of the boundary \(\Gamma\) (denote it by \(z_0=\zeta_0\)) lies on the positive part of the real axis. We now consider a one-parameter family of domains \(B(t)\), obtained from the \((z)\)-plane by making the cut \(\mathcal L(t)+l\), first along the real axis from \(+\infty\) to the point \(z_0\), and then along the curve \(\Gamma\) to the point \(z=\zeta(t)\). As \(t\to T\), the family \(B(t)\) converges to the domain \(B\) as to a kernel. Let the corner points \(z_i\), beginning with \(z_0\), be numbered so that a point with a larger index follows a point with a smaller index under positive traversal of the domain \(B\) along the boundary \(\Gamma\). Suppose, further, that the function \(z=\Phi(w,t_k)\), conformally mapping the half-plane \(\operatorname{Im} w>0\) onto the domain \(B(t_k)\), is known. Then the preimages \(a_{p0}^{(k)}\), \(p=0,1,2,\ldots,2k-1\), of the corner points of the boundary of the domain \(B(t_k)\), and the accessory constants \(M_{p0}^{(k)}\), \(p=0,1,2,\ldots,2k-1\), of the Schwarz derivative \(S(w,t_k)\) are also known. We now extend the cut \(\mathcal L(t_{k+1})+l\) to the next corner point of the curve \(\Gamma\). Then the preimages \(a_p^{(k+1)}(t_{k+1})\), \(p=0,1,2,\ldots,2k+1\), of the corner points of the curve \(\mathcal L(t_{k+1})+l\) under the mapping \(z=\Phi(w,t_{k+1})\), and also the accessory constants \(M_p^{(k+1)}(t_{k+1})\), \(p=0,1,2,\ldots,2k+1\), of the Schwarz derivative \(S(w,t_{k+1})\) are determined as integrals of the system (3) from (1), p. 13, satisfying the initial conditions \(a_p^{(k+1)}(t_k)=a_{p0}^{(k)}\), \(p=2,3,\ldots,2k\); \(a_j^{(k+1)}(t_k)=a_{00}^{(k)}\), \(j=0,1,2k+1\); \(M_p^{(k+1)}(t_k)=M_{p0}^{(k)}\), \(p=2,3,\ldots,2k\);

\[ \lim_{t\to t_k+0}(t-t_k)M_j^{k+1}(t),\qquad j=0,1,2k+1. \]

The system (3) from (1) is integrated by some numerical method (2) from the value of the parameter \(t=t_k\) to the value \(t=t_{k+1}\). The latter, in turn, is determined from the relation

\[ s(t_{k+1})= \int_{t_0}^{t_{k+1}} \left\{ \frac{1}{4}|\gamma(t)|^{-3} |a_0^{(k+1)}(t)-\beta(t)|^3 \exp\left[-2\int\left(\frac{d\gamma}{dt}\right)^2dt\right] \times \right. \]

\[ \left. \times \exp\frac{1}{\pi}\int_{t_0}^{t} \operatorname{Re}H^*(a_0^{(k+1)}(t),t,\tau)\,d\theta(\tau) \right\} 2\left(\frac{d\gamma}{dt}\right)^2dt . \]

For the start of the computation one may use the series

\[ a_p^{(k+1)}(t)=a_{p0}^{(k)}+a_{p1}\sqrt{t-t_k}+a_{p2}\left(\sqrt{t-t_k}\right)^2+\ldots,\quad p=0,1,2,\ldots,2k+1; \]

\[ M_p^{(k+1)}(t)=M_{p0}^{(k)}+m_{p1}\sqrt{t-t_k}+m_{p2}\left(\sqrt{t-t_k}\right)^2+\ldots, \]

\[ p=2,3,\ldots,2k; \]

\[ M_j^{(k+1)}(t)=\frac{m_{j,-1}}{\sqrt{t-t_k}}+m_{j0}+m_{j1}\sqrt{t-t_k}+\ldots;\quad j=0,1,2k+1, \]

in which the functions \(a_p^{(k+1)}(t)\), \(M_p^{(k+1)}(t)\) are expanded. Thus, the constants \(a_p^{(k+1)}(t_{k+1})\) and \(M_p^{(k+1)}(t_{k+1})\) will be determined with the accuracy allowed by the numerical method. Noting further that the function \(z=\Phi(w,t_0)\), mapping the half-plane \(\operatorname{Im} w>0\) onto the plane with a slit along the ray \(l: z_0 \le z < \infty\), is known, and that the family of functions \(\Phi(w,t)\), as \(t\to T\), converges uniformly inside the domain \(\operatorname{Im} w>0\) to \(\Phi(w)\), we conclude that, by the method described above, all the parameters entering Schwarz’s equation \((3)\) for the function \(z=\Phi(w)\) can be determined.

Novosibirsk Electrotechnical
Institute

Received
17 III 1966

CITED LITERATURE

\(^{1}\) B. G. Baibarin, Reports of the II Siberian Conference on Mathematics and Mechanics, 1962.
\(^{2}\) I. S. Berezin, N. P. Zhidkov, Computational Methods, 2, Moscow, 1960.
\(^{3}\) V. Koppenfels, F. Stalman, Practice of Conformal Mappings, IL, 1963.