Construction of complex potentials for multiply connected domains

TL;DR

This paper develops a generalized method to construct complex potentials in multiply connected domains, enabling the mapping of such domains onto slit planes with a focus on computational simplicity.

Contribution

It introduces a generalized approach to construct complex potentials in multiply connected domains, extending previous methods to more complex geometries.

Findings

The method effectively constructs complex potentials with explicit formulas.

It maps multiply connected domains onto slit planes.

The approach is computationally straightforward.

Abstract

The method of reduction of a Fredholm integral equation to the linear system is generalized to construction of a complex potential --- an analytic function in an infinite multiply connected domain with a simple pole at infinity which maps the domain onto a plane with horizontal slits. We consider a locally sourceless, locally irrotational flow on an arbitrary given $n$-connected infinite domain with impermeable boundary. The complex potential has the form of a Cauchy integral with one linear and $n$ logarithmic summands. The method is easily computable.

Equations

$$L_s=\{z=z_s(t),z_s(0)=z_s(2\pi ),t\in [0,2\pi ]\},s=1,\dots ,n.$$

$$z_s(t)=\sum _{k=-m_s}^{n_s}{d}_{ks}{e}^{ikt},t\in [0,2\pi ],s=1,\dots ,n.$$

$$\psi (z_s(t))=\sum _{\sigma =1}^n\frac{1}{\pi i}\underset{0}{\overset{2\pi }{\int }}\psi (z_{\sigma }(\tau ))[\log (z_{\sigma }(\tau )-z_s(t)){]}_{\tau }^{\prime }d\tau .$$

$$q_s(t)=-\mathrm{Im}[\overline{v}{z}_s(t)+\sum _{\sigma =1}^n{c}_{\sigma }\log (z_s(t)-z_{\sigma }^{\ast })]+C_s,$$

$$p_s(t)=\sum _{\sigma =1}^n\frac{1}{\pi }\underset{0}{\overset{2\pi }{\int }}{p}_{\sigma }(\tau )[\arg (z_{\sigma }(\tau )-z_s(t)){]}_{\tau }^{\prime }d\tau +$$

$$+\sum _{\sigma =1}^n\frac{1}{\pi }\underset{0}{\overset{2\pi }{\int }}{q}_{\sigma }(\tau )[\log |z_{\sigma }(\tau )-z_s(t)|{]}_{\tau }^{\prime }d\tau .$$

$$p_s^{\prime }(t)=\sum _{\sigma =1}^n\frac{1}{\pi }\underset{0}{\overset{2\pi }{\int }}{p}_{\sigma }^{\prime }(\tau )K_{\sigma ,s}(\tau ,t)d\tau +Q_s(t),$$

$$K_{\sigma s}(\tau ,t)=-[\arg (z_{\sigma }(\tau )-z_s(t)){]}_t^{\prime },L_{j,s}(\tau ,t)=[\log |z_j(\tau )-z_s(t)|{]}_t^{\prime },$$

$$Q_s(t)=\sum _{\sigma =1}^n\frac{1}{\pi }\underset{0}{\overset{2\pi }{\int }}{q}_{\sigma }^{\prime }(\tau )L_{\sigma ,s}(\tau ,t)d\tau ,$$

$$(\log |z_s(\tau )-z_s(t)|{)}_t^{\prime }=\mathrm{Re}{\left(\log \sum _{k=-m_s}^{n_s}{d}_{ks}[e^{ik\tau }-e^{ikt}]\right)}_t^{\prime }=\mathrm{Re}(\log \sin \frac{\tau -t}{2}+$$

$${+\log \left[\sum _{k=1}^{n_s}{d}_{ks}{e}^{ikt}\sum _{l=0}^{k-1}{e}^{il(\tau -t)}-\sum _{k=1}^{m_s}{d}_{(-k)s}{e}^{-ik\tau }\sum _{l=1}^{k-1}{e}^{il(\tau -t)}\right])}_t^{\prime }=$$

$$=-\frac{1}{2}\cot \frac{\tau -t}{2}+{\left(\log \left|\sum _{k=1}^{n_s}{d}_{ks}{e}^{ikt}\sum _{l=0}^{k-1}{e}^{il(\tau -t)}-\sum _{k=1}^{m_s}{d}_{(-k)s}{e}^{-ik\tau }\sum _{l=1}^{k-1}{e}^{il(\tau -t)}\right|\right)}_t^{\prime }.$$

$$\frac{1}{\pi }\underset{0}{\overset{2\pi }{\int }}[\log |z_{\sigma }(\tau )|{]}^{\prime }\cot \frac{\tau -t}{2}d\tau$$

$$\left(\begin{array}[]{ccccccc}I-K_{1,1}&-K_{2,1}&\ldots&-K_{n,1}\\ -K_{1,2}&I-K_{2,2}&\ldots&-K_{n,2}\\ \ldots&\ldots&\ldots&\ldots\\ -K_{1,n}&-K_{2,n}&\ldots&I-K_{n,n}\end{array}\right)\left(\begin{array}[]{c}p_{1}^{\prime}\\ \vdots\\ p_{n}^{\prime}\end{array}\right)=$$

$$\displaystyle=\left(\begin{array}[]{c}Q_{1}\\ \vdots\\ Q_{n}\end{array}\right).$$

$$p_s^{\prime }(t)=\sum _{l=1}^M{\alpha }_{ls}\cos lt+{\beta }_{ls}\sin lt,s=1,\dots ,n.$$

$$\left(\begin{array}{ccccc}{A}_{11}& B_{11}& A_{12}& \dots & B_{1n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ C_{n1}& D_{n1}& C_{n2}& \dots & D_{nn}\end{array}\right)\times$$

$$\times \left(\begin{array}{c}{\alpha }_1\\ {\beta }_1\\ {\alpha }_2\\ {\beta }_2\\ ⋮\\ {\beta }_n\end{array}\right)=\left(\begin{array}{c}{a}_1\\ b_1\\ a_2\\ b_2\\ ⋮\\ b_n\end{array}\right),$$

$$a_{js}=\frac{1}{\pi }\underset{0}{\overset{2\pi }{\int }}{Q}_s(t)\cos jtdt,b_{js}=\frac{1}{\pi }\underset{0}{\overset{2\pi }{\int }}{Q}_s(t)\sin jtdt,j=1,\dots ,n,s=1,\dots ,n,$$

$$A_{\sigma sjk}={\delta }_{\sigma s}{\delta }_{jk}-\frac{1}{{\pi }^2}\underset{0}{\overset{2\pi }{\int }}\cos k\tau d\tau \underset{0}{\overset{2\pi }{\int }}{K}_{\sigma s}(\tau ,t)\cos jtdt,$$

$$B_{\sigma sjk}=-\frac{1}{{\pi }^2}\underset{0}{\overset{2\pi }{\int }}\sin k\tau d\tau \underset{0}{\overset{2\pi }{\int }}{K}_{\sigma s}(\tau ,t)\cos jtdt,$$

$$C_{\sigma sjk}=-\frac{1}{{\pi }^2}\underset{0}{\overset{2\pi }{\int }}\cos k\tau d\tau \underset{0}{\overset{2\pi }{\int }}{K}_{\sigma s}(\tau ,t)\sin jtdt,$$

$$D_{\sigma sjk}={\delta }_{\sigma s}{\delta }_{jk}-\frac{1}{{\pi }^2}\underset{0}{\overset{2\pi }{\int }}\sin k\tau d\tau \underset{0}{\overset{2\pi }{\int }}{K}_{\sigma s}(\tau ,t)\sin jtdt,$$

$$\tilde{Y}=\tilde{P}\tilde{Y}+\tilde{Q},$$

$$\left(\begin{array}{cc}{I}_F-P_F& S\\ R& I_{\infty }-V\end{array}\right)\left(\begin{array}{c}{Y}_1\\ Y_2\end{array}\right)=\left(\begin{array}{c}{Q}_1\\ Q_2\end{array}\right).$$

$$(I_F-P_F)[I_F-(I_F-P_F{)}^{-1}S(I_{\infty }-V{)}^{-1}R]Y_1=Q_1-S(I_{\infty }-V{)}^{-1}{Q}_2.$$

$$\Vert Y_1-{\tilde{Y}}_1\Vert \le \frac{1}{1-r}\Vert (I_F-P_F{)}^{-1}\Vert \Vert S(I_{\infty }-V{)}^{-1}\Vert \Vert Q_2\Vert +\frac{r}{1-r}\Vert (I_F-P_F{)}^{-1}\Vert \Vert Q_1\Vert .$$

$${\psi }_s(t)={\tilde{p}}_s(t)+i{q}_s(t)+q_{0s},{\tilde{p}}_s(t)=\sum _{l=1}^M\frac{{\alpha }_{ls}}{l}\sin lt-\frac{{\beta }_{ls}}{l}\cos lt,s=1,\dots ,n.$$

$$\sum _{s=1}^n\frac{1}{2\pi i}\underset{0}{\overset{2\pi }{\int }}\left[{\tilde{p}}_s(\tau )+i{q}_s(\tau )+q_{0s}\right][\log (z_s(\tau )-z_k^{\ast }){]}_{\tau }^{\prime }d\tau =0,k=2,\dots ,n.$$

$$\varphi (z)=\sum _{s=1}^n\frac{1}{2\pi i}\underset{0}{\overset{2\pi i}{\int }}\frac{({\tilde{p}}_s(\theta )+i{q}_s(\theta )+q_{0s})z_s^{\prime }(\theta )}{z_s(\theta )-z}d\theta +\overline{v}z+\sum _{s=1}^n{c}_s\log (z-z_s^{\ast }).$$

Full text

Keywords: potential flow; multiply connected domain; Fredholm integral equation.

Introduction

Conformal mappings by the analytical functions of complex variable play an important part in solution of many problems of mechanics and mathematics, particularly in the case of plane potential fields and Laplace equation solution [1]. It is known that a complex potential of a plane locally sourceless, locally irrotational steady flow in any $n$-connected infinite domain may have only logarithmic summands to the analytic function with the given simple pole at infinity [2]. Here we present the method of construction of a complex potential for this flow in any multiply connected domain with impermeable boundary through approximate solution of the Fredholm equation. We give the following formulation of the problem: given the velocity at the infinity and the circulations around the contours we are able to find the analytic function with constant imaginary parts at the domain boundary and the given simple pole at infinity. We prove convergence of the method and give certain clarifying examples of its application. We base our construction of the complex potential in an infinite multiply-connected domain on the approximation of the analytic mapping of our domain on the domain with horizontal slits with the given simple pole at infinity.

The solution of the integral equations in our method is reduced to the solution of an infinite linear system. We obtain an approximate mapping by solving the finite system with a truncated matrix.

The method computational complexity equals $O(M^{3})$, here $M$ is the order of the corresponding system.

Complex potential construction for the flow around several contours by means of Cauchy integral

Consider an infinite $n$-connected domain $D_{z}$ bounded by the simple smooth curves $L_{s}$ given by the equations

[TABLE]

We also assume that the boundary curves $L_{s}$ complex representations are as follows:

[TABLE]

The parametrization traces the contours $L_{s}$, $s=1,\ldots,n$, counterclockwise.

Note that the flow may include circulations around certain connected components of the domain.

Theorem 1**.**

A complex potential $f(z)$ of the flow in any $n$-connected domain $D_{z}$ can be approximately constucted by reduction to solution of a linear system.

Proof.

Assume that $v\in\mathbb{C}$ is the velocity of the flow at infinity. Consider the complex potential in the form of the function $\phi(z)=\overline{v}z+\sum\limits_{s=1}^{n}c_{s}\log(z-z^{*}_{s})+\psi(z)$ [2]. Here $z^{*}_{s}$ is a point inside the domain bounded by the contour $L_{s}$, $c_{s}=\frac{\Gamma_{s}}{2\pi i}$, $\Gamma_{s}$ is the circulation around the contour $L_{s}$, $s=1,\ldots,n$, [2], and $\psi(z)$ is the unknown analytical in $D_{z}$ function.

Any contour $L_{s}$, $s=1,\ldots,n$, is a part of a flow line for this complex potential, so the imaginary part of the function $\phi(z)$ is constant on these contours.

According to [7] the necessary and sufficient condition for $\psi(t)$, to be analytic in $D_{z}$ are the boundary relations

[TABLE]

where $t\in[0,2\pi]$, $s=1,\ldots,n$.

We introduce the new functions $p_{s}(t)$, $q_{s}(t)$ as follows: $\psi(z_{s}(t))=p_{s}(t)+iq_{s}(t)$, $s=1,\ldots,n$. Note that

[TABLE]

here $C_{s}$ is constant for any $s=1,\ldots,n$.

We separate the real part of both sides of equation (1):

[TABLE]

After differentiating relation (2) with respect to $t$ and integrating the results by parts, we obtain the following relations on the functions $p_{s}^{\prime}(t)$:

[TABLE]

where

[TABLE]

[TABLE]

here by formula (2) $q^{\prime}_{s}(t)=-{\rm Im}[\overline{v}z_{s}^{\prime}(t)+\sum\limits_{\sigma=1}^{n}\frac{c_{\sigma}z_{s}^{\prime}(t)}{z_{s}(t)-z^{*}_{\sigma}}]$, $s=1,\ldots,n$.

The kernel $L_{\sigma,s}$ has a singularity in the form of $\cot\frac{\tau-t}{2}$ for $\sigma=s$:

[TABLE]

[TABLE]

[TABLE]

The Cauchy principal value integral

[TABLE]

can be calculated via Hilbert formula [10] as in [9].

Finally we obtain the following system of Fredholm integral equations of the second kind which can be written in the operator form as follows:

[TABLE]

[TABLE]

The last operator system can be reduced to the infinite linear system over the Fourier coefficients of the unknown functions $p_{s}^{\prime}(t)$, $s=1,\ldots,n$, if we find the coefficients of double Fourier expansions of the kernels of integral operators and compare the coefficients with the same trigonometric functions. Approximate solution of the infinite system over Fourier coefficients of the unknown functions is a solution of a truncated system over the Fourier coefficients of the unknown functions.

Existence of the exact solution of system (5) and convergence of the approximate solution to the exact one provided $M\to\infty$ were proved in [8] for the case of conformal mapping of a simply connected domain. This proof can be applied to the case of multiply connected domain if we replace the corresponding space $l^{2}$ by the space $l^{2}\times l^{2}\times\cdots\times l^{2}$.

We search for the approximate solution of system (5) in the form of Fourier polynomials:

[TABLE]

Now integral Fredholm equations of the second kind in (5) can be reduced to the linear system over Fourier coefficients $\alpha_{ls}$ and $\beta_{ls}$, $s=1,\ldots,n$:

[TABLE]

[TABLE]

where $\alpha_{s}=(\alpha_{1s},\ldots,\alpha_{ns})^{T}$, $\beta_{s}=(\beta_{1s},\ldots,\beta_{ns})^{T}$. The vectors $a_{s}=(a_{1s},\ldots,a_{ns})^{T}$, $b_{s}=(b_{1s},\ldots,b_{ns})^{T}$ on the right-hand side of the system consist of the elements

[TABLE]

The block matrices $A_{\sigma s}$, $B_{\sigma s}$, $C_{\sigma s}$, $D_{\sigma s}$, $\sigma,s=1,\ldots,n$, of size $M\times M$ consist of the elements

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $j,k=1,\ldots,n$, $\delta_{rt}$ is the Kronecker delta function.

Now this system can be reduced to the infinite equation system over the unknown coefficients $\alpha_{k,j}$, $\beta_{k,j}$, $j=0,\ldots,n$, $k=1,\ldots,\infty$, in the equivalent form

[TABLE]

where $\tilde{Y}=(\alpha_{1,0},\alpha_{1,1},\ldots,\alpha_{1,n},\beta_{1,0},\beta_{1,1},\ldots,\beta_{1,n},\alpha_{2,0},\ldots)\in l_{2}$, the infinite matrix $\tilde{P}$ consists of the elements $\frac{1}{\pi^{2}}\int\limits_{0}^{2\pi}f(mt)dt\int\limits_{0}^{2\pi}g(p\tau)(\arg[z_{k}(\tau)-z_{j}(t)])^{\prime}_{\tau}d\tau$ or $\frac{1}{\pi^{2}}\int\limits_{0}^{2\pi}f(mt)dt\int\limits_{0}^{2\pi}g(p\tau)(\ln|z_{k}(\tau)-z_{j}(t)|)^{\prime}_{\tau}d\tau$ with $f(x)$, $g(x)$ equal to $\cos x$ or $\sin x$. These elements being the Fourier coefficients of double Fourier series of $(\mathrm{arg}(z_{k}(\tau)-z_{j}(t)))^{\prime}_{\tau}$ or $\ln|z_{k}(\tau)-z_{j}(t)|^{\prime}_{\tau}$, $\tilde{Q}$ being the sequence of the corresponding Fourier coefficients of the functions $q_{j}(t)$, $j=1,\ldots,n$.

We need to construct the approximate solution $\tilde{y}_{j}(t)$ of equation system (11) in the trigonometric polynomial form $\tilde{y}_{j}(t)=\sum\limits_{k=1}^{M}\alpha_{k,j}\cos kt+\beta_{k,j}\sin kt$ in order to apply truncated linear system as in [16]. So we have to find the vector $\tilde{Y}_{F}$ with zero coordinates starting with the $(2M(n+1)+1)$-th one which approximates the infinite vector $\tilde{Y}$. Further on we identify the vector-function $Y$, the integral operator $P$, the vector-function $Q$ with the sequence $\tilde{Y}$, the infinite matrix $\tilde{P}$ and the sequence $\tilde{Q}$, respectively.

Evidently the kernels $(\mathrm{arg}(z_{j}(\tau)-z_{k}(t)))^{\prime}_{\tau}$ of system (9) are infinitely differentiable for $k\not=j$. Due to Cauchy theorem we have $\mathrm{arg}(z_{j}(\tau)-z_{j}(t))=$ $\mathrm{arg}z^{\prime}(t+\theta(\tau-t))$ with $\theta\in(0,1)$, so this function is well-defined for $t,\tau\in[0,2\pi]$ and $\lim\limits_{\tau\to t}\mathrm{arg}(z_{j}(\tau)-z_{j}(t))^{\prime}_{\tau}=$ $k(t)|z^{\prime}(t)|/2$ where $k(t)$ is the curvature of the boundary curve at the corresponding point. It can be easily verified that the kernel $(\mathrm{arg}(z_{j}(\tau)-z_{j}(t)))^{\prime}_{\tau}$ is at least twice differentiable with respect to both variables. So the double complex Fourier coefficients of $(\mathrm{arg}(z_{k}(\tau)-z_{j}(t)))^{\prime}_{\tau}$ have the following estimates: $|c_{k,j,l,t}|<\frac{U}{|l|^{2}|t|^{2}}$.

For $F=(n+1)2M$ integral equation system (9) reduces to infinite linear system (11) which can be presented as follows:

[TABLE]

Here $P_{F}$ is an $F\times F$ block matrix $P=\left(\begin{matrix}P_{1,1}&\ldots&P_{1,n}\\ \ldots&\ldots&\ldots\\ P_{n,1}&\ldots&P_{n,n}\end{matrix}\right)$, $M\times M$ matrices $P_{j,k}$ correspond to integral summands of (9), $j,k=0,\ldots,n$, $S$ is an $F\times\infty$ matrix, $R$ is an $\infty\times F$ matrix, $V$ is an $\infty\times\infty$ matrix, $I_{F}$ and $I_{\infty}$ are the identity matrices of relative sizes. Each of the vectors $Q_{1}$ and $Y_{1}$ has $F$ coordinates, the vectors $Q_{2}$ and $Y_{2}$ have the infinite number of coordinates. The Fourier coefficients of the smooth functions tend to zero as their numbers tend to infinity, so the coefficients of the matrices $S,R$ and $V$ together with the coordinates of $Q_{2}$ decrease rapidly as $F\to\infty$. Due to Theorem assumptions and the Fourier coefficients speed of convergence to zero the matrix norm of $V$ and the vector norm of $Y_{2}$ tend to zero as $F\to\infty$.

Let us prove that there exists the number $T\in\mathbb{N}$ such that the matrix operator $I_{F}-P_{F}$ is invertible $\forall F>T$ since the limit for $P_{F}$ integral operator $P$ is compact and the operator $I-P$ is invertible due to the lemma assumption. Note that we do not distinguish a finitely dimensional vector and the Fourier polynomial with the corresponding finite set of coordinates in our proof. Recall first that due to chapter VI, paragraph 1 of [17] $\|P-P_{F}\|\to 0$ if $F\to\infty$. The operator norm that we deal with here is the usual operator norm for the Hilbert space mappings. Let us assume that $\forall T\in\mathbb{N}$ there exists $s_{l}>T$ such that the spectrum of $P_{s_{l}}$ contains $1$. Then there exists an infinite sequence $({v}_{s_{l}})_{l\in\mathbb{N}}\subset L^{2}$ such that $\|{v}_{s_{l}}\|=1$ and $P_{s_{l}}{v}_{s_{l}}={v}_{s_{l}}$. Let us prove that then there should exist at least one limit point for the sequence $\{{v}_{s_{l}}\}_{l\in\mathbb{N}}$. Since the operator $P$ is compact there exist both a subsequence $\{{v}_{s_{k_{j}}}\}_{j\in\mathbb{N}}$ and an element ${w}_{0}\in L^{2}$ so that $P{v}_{s_{k_{j}}}\to{w}_{0}$, $(j\to\infty)$. Then $\|P_{s_{k_{j}}}{v}_{s_{k_{j}}}-{w}_{0}\|=\|P_{s_{k_{j}}}{v}_{s_{k_{j}}}-P{v}_{s_{k_{j}}}+P{v}_{s_{k_{j}}}-{w}_{0}\|\leq\|P_{s_{k_{j}}}{v}_{s_{k_{j}}}-P{v}_{s_{k_{j}}}\|+\|P{v}_{s_{k_{j}}}-{w}_{0}\|\leq\|P_{s_{k_{j}}}-P\|+\|P{v}_{s_{k_{j}}}-{w}_{0}\|\to 0$, $(j\to\infty)$. Thus $\|{v}_{s_{k_{j}}}-{w}_{0}\|=\|P_{s_{k_{j}}}{v}_{s_{k_{j}}}-{w}_{0}\|\to 0$, $(j\to\infty)$. Hence ${v}_{s_{k_{j}}}\to{w}_{0}$, $(j\to\infty)$. Note that since $\|{v}_{s_{k_{j}}}\|=1$, $\forall j\in\mathbb{N}$, the element ${w}_{0}$ is nondegenerate. Let us show now that the relation $P{w}_{0}={w}_{0}$ holds true. Indeed, we have $\|P{w}_{0}-{w}_{0}\|=\|P{w}_{0}+P{v}_{s_{k_{j}}}-P{v}_{s_{k_{j}}}-{w}_{0}\|\leq\|P\|\|{w}_{0}-{v}_{s_{k_{j}}}\|+\|P{v}_{s_{k_{j}}}+P_{s_{k_{j}}}{v}_{s_{k_{j}}}-P_{s_{k_{j}}}{v}_{s_{k_{j}}}-{w}_{0}\|\leq\|P\|\|{w}_{0}-{v}_{s_{k_{j}}}\|+\|P_{s_{k_{j}}}{v}_{s_{k_{j}}}-P{v}_{s_{k_{j}}}\|+\|{w}_{0}-P_{s_{k_{j}}}{v}_{s_{k_{j}}}\|\leq\|P\|\|{w}_{0}+{v}_{s_{k_{j}}}\|+\|P-P_{s_{k_{j}}}\|+\|{w}_{0}-{v}_{s_{k_{j}}}\|\to 0$, $(j\to\infty)$. Hence the spectrum of $P$ contains $1$. A contradiction with one of the assumptions.

We now take the number $F$ so that $\|V\|<1$ and the matrix $I_{F}-P_{F}$ possesses the inverse one. Now we have the relation

[TABLE]

Obviously one can choose the value of $F$ so large that $\|S(I_{\infty}-V)^{-1}R\|=O(1/F^{2})\leq r$ where $r<1$ is arbitrary small. Now we estimate the norm of the difference between the solution $x_{1}$ and the solution $\tilde{x}_{1}$ of the truncated system $(I_{F}-P_{F})\tilde{x}_{1}=y_{1}$:

[TABLE]

Consider the first summand on the right-hand side of the last inequality. Recall the Jackson’s inequality: if ${\displaystyle f:[0,2\pi]\to\mathbb{C}}$ is an $r$ times differentiable periodic function such that ${\displaystyle|f^{(r)}(x)|\leq 1,\quad 0\leq x\leq 2\pi,}$ then, for every positive integer $n$, there exists a trigonometric polynomial${\displaystyle T_{n-1}}$ of degree at most ${\displaystyle n-1}$ such that $|f(x)-T_{n-1}(x)|\leq\frac{C(r)}{n^{r}}$ for any $x\in[0,2\pi]$ where ${\displaystyle C(r)}$ depends only on $r$ [18]. So the vector norm of $y_{2}$ can be estimated by this inequality by $K/F^{2}$. The second summand also behaves no better than $O(1/F^{2})$. So the error due to the series tail is $O(1/F^{2})$.

The functions $\psi_{s}(t)$, $s=1,\ldots,n$, can be restored via the derivatives $p^{\prime}_{s}(t)$ given by formula (10) and the functions $q_{s}(t)$ with arbitrary constant summands $q_{0s}$

[TABLE]

We choose the constant summand $q_{01}=0$.

We obtain the values of the other constant summands $q_{0s}$, $s=2,\ldots,n$, in the following way. We take $n-1$ points $z_{k}^{*}$ in $n-1$ finite components bounded by $L_{k}$, $k=2,\ldots,n$. The functions $\psi_{s}(t)$, $s=1,\ldots,n$, are the boundary values of the analytical in $D_{z}$ function, so the Cauchy integral with the corresponding density along the boundary of $D_{z}$ vanishes at the points $z_{k}^{*}$, $k=2,\ldots,n$. Therefore we have the linear complex system

[TABLE]

over the unknown complex numbers $q_{0s}$, $s=2,\ldots,n$.

Now we have the functions $\psi_{s}(t)$, $s=1,\ldots,n$, $t\in[0,2\pi]$, and therefore we can restore the complex potential.

The approximate analytical potential now has the form of the Cauchy integral

[TABLE]

∎

If the given contour $L_{s}$, $s\in\{1,\ldots,n\}$, possesses a cusp then the flow is steady if its branching point for $L_{s}$ coincides with the cusp. In order to put the branching points of the flow at the cusps of the given contours so that the velocity there vanishes we have to modify the circulations $\Gamma_{k}$ around the contours $L_{k}$, $k=1,\ldots,n$.

3. Examples

1. Consider two elliptic holes with the boundaries $\exp\left(\frac{1}{4}(-i)\pi\right)(3\exp(it)-\exp(-it))+10i$ and $2\exp(-it)+4\exp(it)$, $t\in[0,2\pi]$, with velocity at infinity equal to $1+0.1i$ (Fig. 1). The circulations $\Gamma_{1}=\Gamma_{2}=0$. We give the flow lines on Fig. 1 (a). Fig. 1 (b) shows the absolute values of velocity. The minimal velocity values happen at the flow branching points.

2. Consider two elliptic holes bounded by $\exp\left(\frac{1}{4}(-i)\pi\right)(3\exp(it)-\exp(-it))+10i$ and $2\exp(-it)+4\exp(it)$, $t\in[0,2\pi]$, with velocity at infinity equal to $1+0.1i$. In order to shift the branching points of the flow at the contours we introduce the circulations $-0.4\pi$ and $-1.2\pi$ around the lower and the upper contours, respectively. We give the flow lines on Fig. 2 (a). Fig. 2 (b) shows the absolute values of velocity. The flow branching points on the lower contour shift downwards from that of item 1.

3. Consider two elliptic holes bounded by $\exp\left(\frac{1}{4}(-i)\pi\right)(3\exp(it)-\exp(-it))+10i$ and $2\exp(-it)+4\exp(it)$, $t\in[0,2\pi]$, with velocity at infinity equal to $1+0.1i$. In order to shift the branching points of the flow at the contours we introduce the circulations $0.4\pi$ and $-1.2\pi$ around the lower and the upper contours, respectively. We give the flow lines on Fig. 3 (a). Fig. 3 (b) shows the absolute values of velocity. The difference in placement of the flow branching points compared to the previous example is clear at the upper contour.

4. Consider two noncircular domains with the cusp points $2\left(0.15e^{2it}+1.6e^{it}+\left(0.3i+0.8\right)e^{-it}-0.25i{e}^{-2it}\right)$ and $0.15e^{2it}+1.6e^{it}+\left(0.3i+0.8\right)e^{-it}-0.25i{e}^{-2it}+10i$, $t\in[0,2\pi]$. We construct the flow with velocity at infinity equal to $1+0.1i$. We give the flow lines on Fig. 4 (a). Fig. 4 (b) shows the absolute values of velocity.

5. Consider two noncircular domains with the cusp points $2\left(0.15e^{2it}+1.6e^{it}+\left(0.3i+0.8\right)e^{-it}-0.25i{e}^{-2it}\right)$ and $\left(\frac{i}{2}+\frac{\sqrt{3}}{2}\right)\left(0.15e^{2it}+1.6e^{it}+\left(0.3i+0.8\right)e^{-it}-0.25i{e}^{-2it}\right)+10i$, $t\in[0,2\pi]$. We put the flow velocity at infinity equal to $1+0.1i$. We give the flow lines on Fig. 5 (a). Fig. 5 (b) shows the absolute values of velocity.

Consider different possible mutual positioning of the holes from Example 5: $2\left(0.15e^{2it}+1.6e^{it}+\left(0.3i+0.8\right)e^{-it}-0.25i{e}^{-2it}\right)$ and $\left(\frac{i}{2}+\frac{\sqrt{3}}{2}\right)\left(0.15e^{2it}+1.6e^{it}+\left(0.3i+0.8\right)e^{-it}-0.25i{e}^{-2it}\right)+5i$. Again we put the flow velocity at infinity equal to $1+0.1i$. We give the flow lines on Fig. 6 (a). Fig. 6 (b) shows the absolute values of velocity. Note the large high velocity zone between the contours.