# One Parameter Families of Conformal Mappings
of Bounded Doubly Connected Polygonal Domains

A. Dyutin  Kazan Federal University, Kremlyovskaya str. 35, Tatarstan, 420008,
Russian Federation<EMAIL_ADDRESS>and  S. Nasyrov  Kazan Federal
University, Kremlyovskaya str. 35, Tatarstan, 420008, Russian Federation
<EMAIL_ADDRESS>
###### Abstract.

We suggest an approximate method of finding a conformal mapping of an annulus
onto an arbitrary bounded doubly connected polygonal domain. The method is
based on the parametric Loewner–Komatu method. We consider smooth one
parameter families $\mathcal{F}(z,t)$ of conformal mappings of concentric
annuli onto doubly connected polygonal domains $\mathcal{D}(t)$ which are
obtained from a fixed doubly connected polygonal domain $\mathcal{D}$ by
making a finite number of rectilinear slits of variable lengths; meanwhile we
do not require the family of domains $\mathcal{D}(t)$ to be monotonous. The
integral representation for the conformal mappings $\mathcal{F}(z,t)$ has
unknown (accessory) parameters. We find a PDE for this family and then deduce
a system of PDEs describing dynamics of accessory parameters and conformal
modulus of $\mathcal{D}(t)$ when changing the parameter $t$. We note that the
right-hand sides of equations in the system of ODEs involve functions standing
for speeds of movement of the end points of the slits. This allows us to
control completely dynamics of the slits and to seek their agreed change, if
we have more than one slit. We also give some results of numerical
calculations which show the efficiency of the suggested method.

###### Key words and phrases:

Doubly connected domain, polygonal domain, conformal modulus, conformal
mapping, Schwarz–Christoffel formula, accessory parameters, one parameter
family of functions, parametric method, elliptic functions, elliptic
integrals, Loewner–Komatu equation.

##  1\. Introduction

In geometric function theory and applications, conformal mappings of polygonal
domains play an important role. In the simply connected case, by the Riemann
mapping theorem, there exists a conformal mapping of a given polygonal domain
onto either the unit disk or the upper half-plane. The inverse mapping is
given by the Schwarz–Christoffel formula (see, e.g. [31]). In the doubly
connected case, we can map a polygonal domain onto a concentric annulus (see,
e.g. [33, ch. 5, § 1]), and the inverse mapping is expressed via a generalized
Schwarz–Christoffel integral, with the help of elliptic and theta functions
(see, e.g. [2, 21]). First the integral representations for such mappings were
obtained by N. Akhiezer [3] (see also [2, ch. 8, §. 48]), later similar
formulas were deduced by G. M. Goluzin [5] and Yu. Komatu [38, 39] (see also
[36, § 17.5]).

In the multiply connected case, we can map a polygonal domain onto a circular
one, i.e. domain bounded with a finite number of disjoint circles (see, e.g.
[33, ch. 5, § 6]). In 2001 T. K. Delillo, A. R. Elcrat, J. A. Pfaltzgraff [27]
found a formula for the conformal mapping of the exterior of a finite number
of disks onto the exterior of polygonal lines; in 2006 they together with T.
A. Driscoll obtained a system of equations to determine unknown parameters in
such representations [30]. D. G. Crowdy [22, 23] also obtained a formula for
conformal mappings of a multiply connected circular domain onto a bounded and
unbounded multiply connected polygonal domain. It is of interest to find
conformal mappings of circular polygons onto canonic domains; we can note that
papers [16, 24, 25, 28, 29] with results obtained in this direction.

An undoubted advantage of the Schwarz–Christoffel formula and its
generalizations is its simplicity. Its essential disadvantage is the fact that
Schwarz–Christoffel integrals contain unknown parameters. First of all, they
are the preimages of the vertices of polygonal lines bounding the domains.
Secondly, the multiply connected case is complicated by the fact that the
conformal parameters of domains (so-called conformal moduli) are also unknown.
In the doubly connected case, we have only one such parameter which we need to
determine, it is called conformal modulus. Now we will recall its definition:
if a doubly connected domain $\mathcal{D}$ is conformally equivalent to an
annulus $\\{\tau\in\mathbb{C}:q<|\tau|<1\\}$, then its conformal modulus is
defined as

$\text{Mod}(\mathcal{D})=\frac{1}{2\pi}\log q^{-1}\,.$

There exist various methods of finding unknown (accessory) parameters in
Schwarz–Christoffel integrals and their generalizations. One of the most
interesting of them is the Loewner parametric method which has other important
applications in the theory of univalent functions (see, e.g., the
Schwarz–Christoffel integrals [19], and the monograph [1]). Using the method,
L. de Branges [20] in 1984 proved the famous Bieberbach conjecture on the
estimates of the moduli of the Taylor coefficients for univalent functions in
the unit disk [17]. We note the recent increase of interest to probabilistic
applications of the Loewner method including the stochastic Loewner equation
or the Schramm–Loewner equation (SLE) (see, e.g. [6, 15, 32, 34]).

The parametric Loewner method is based on considering one parameter families
of univalent functions depending on a real parameter, that map a canonic
domain onto a family of domains which are obtained from a fixed one by making
an increasing slit, and investigation of a differential equation which the
family satisfy to; it is called the Loewner equation.

The method of finding accessory parameters in Schwarz–Christoffel integrals,
based on the Loewner parametric method, in the simply connected case was first
suggested by P. P. Kufarev [10]; using the Loewner equation he showed that we
can determine these parameters by numeric integration of a system of ODEs. In
1960 Yu. V. Chistyakov [14] first fulfilled Kufarev’s method by performing
numerical calculations. In 1979 T. R. Hopkins and D. E. Roberts [37] also
tested this method. Later V. Ya. Gutlyanskii and A. O. Zaidan [7] suggested a
modification of Kufarev’s method. A slightly different approach, close to one
in [7], was suggested by L. Yu. Nizamieva [13]; her study was based on the
using of the Hilbert boundary value problem with piecewise constant
coefficients. A method of finding accessory parameters in the generalized
Schwarz–Christoffel integral, giving a conformal mapping of the half-plane
onto a multi-sheeted polygonal domain with branch-points was suggested in
[11]. In [42], the parametric method was developed for the case of several
slits. Note that the case of several slits of an arbitrary form was considered
in [18].

In the multiply connected case, the use of the parametric method is more
complicated than in the simply connected one. Making a slit in a non-simply
connected domain change its conformal moduli and this puts the domain with a
slit out of the class of domains conformally equivalent to the initial one.
Instead, we consider families of conformal mappings, defined on variable
domains. In the doubly connected case, there is the Loewner-Komatu equation
which is a generalization of the Loewner equation; its development has been
started by Yu. Komatu [39] and G. M. Goluzin [4]. Then M. R. Kuvaev and P. P.
Kufarev [8, 9] extended the method to the general case of multiply connected
domains.

This paper is devoted to finding accessory parameters in the generalized
Schwarz–Christoffel integral with the help of the parametric method in case
when we simultaneously make several rectilinear slits of variable lengths in
the initial bounded doubly connected domain $\mathcal{D}$; we denote the
corresponding family of polygonal domains with slits by $\mathcal{D}(t)$; here
$t$ is a real parameter.

Now we will describe the structure of the paper, the main results and give
some comments.

In § 2 we describe some known results from the theory of elliptic functions
and elliptic integrals.

In § 3 we give an integral representation for the conformal mapping of a
concentric annulus onto a fixed bounded doubly connected polygonal domain
(Theorem 3.1).

Then in § 4 with the use of this representation we consider smooth one
parameter families $\mathcal{F}(z,t)$ of conformal mappings of annuli onto
doubly connected polygonal domains $\mathcal{D}(t)$. We note that at the same
time we make several slits and do not require that the family $\mathcal{D}(t)$
to be monotonous (in the sense of inclusion $\subset$). This allow us to both
increase and decrease the lengths of some slits. We deduce a PDE for the
family of such conformal mappings which is a modification of the
Loewner–Komatu equation for the case of several rectilinear slits with
prescribed dynamics of their endpoints (Theorem 4.1). From this equation we
obtain a system of ODEs describing the dynamics of accessory parameters and
conformal modulus of doubly connected polygonal domains (Theorem 4.2). Solving
the Cauchy problem for the system, we find all unknown parameters in the
integral representation of $\mathcal{F}(z,t)$. We especially note that we can
control the location and velocities of end points of the slits, since their
speeds are contained explicitly in the right-hand sides of the obtained
differential equations.

We note that we use the approach which was applied earlier for one parameter
families of rational functions [41], conformal mappings of complex tori [12],
and for doubly connected domains which are the exterior of two rectilinear
slits of variable lengths [26], while deducing the system of ODEs. The results
obtained in this paper extend the results of [42], where one parametric
families of polygonal domains with several slits where considered, to the
cases of doubly connected domains.

Using the Caratheodory kernel convergence theorem (see, e.g. [33]), we can cut
off domains of a more complicated geometry from a given polygonal domain
$\mathcal{D}$. We can also repeat this process and obtain more complicated
domains in a few steps. In this case, the method could be fulfilled by
consistent solving of several Cauchy problems for systems of ODEs. The values
of accessory parameters and conformal modulus, obtained on every step, except
for the final one, give the initial conditions for the Cauchy problem solved
on the next step.

Finally, in § 5 we show the efficiency of the suggested method by giving the
results of numeric calculations. We take a rectangle with a slit on the axis
of symmetry as an initial domain $\mathcal{D}$. To find the initial values of
parameters we use the formulas obtained in § 3 (Example 3.1). Then we not only
move the rectilinear slit along the axis of symmetry (Example 5.1), but also
cut off polygonal domains moving the interior component of the boundary of the
polygonal domain (Examples 5.2 and 5.3). We note that the accuracy of
calculations for the preimages of the vertices is about $10^{-9}$ and for the
lengths of the sides of boundary polygonal lines is of order $10^{-6}$.

##  2\. Preliminary information

Elliptic integrals and elliptic functions. First we will recall some facts
from the theory of elliptic integrals and elliptic functions (see, e.g. [2,
21, 40]).

A meromorphic function in $\mathbb{C}$ is called elliptic, if it has periods
$\omega_{1}$ and $\omega_{2}$, linearly independent over $\mathbb{R}$. Let us
fix an arbitrary point $z_{0}\in\mathbb{C}$ and consider the parallelogram
with vertices at the points $z_{0}$, $z_{0}+\omega_{1}$,
$z_{0}+\omega_{1}+\omega_{2}$, $z_{0}+\omega_{2}$; it is called the period
parallelogram. We add to this parallelogram all points of the form
$z_{0}+r_{1}\omega_{1}+r_{2}\omega_{2}$ $(0\leq r_{1}<1,0\leq r_{2}<1)$. For
$z_{0}=0$, the period parallelogram is called basic. The order of a
nonconstant elliptic functions is the number of its poles in the period
parallelogram taking into account multiplicities. Denote by
$\Omega=\\{m_{1}\omega_{1}+m_{2}\omega_{2}\\}$ the lattice generated by the
periods $\omega_{1}$ and $\omega_{2}$ $(m_{1},m_{2}\in\mathbb{Z})$. Further we
will denote by $\omega$ an arbitrary element of the lattice and assume that
$\text{Im}(\omega_{2}/\omega_{1})>0$.

One of the main elliptic functions is the Weierstrass $\wp$–function:

$\wp(z)=\wp(z;\,\omega_{1},\omega_{2})=\frac{1}{z^{2}}+\sum_{\omega\,\in\,\Omega,\,\omega\,\neq\,0}\left(\frac{1}{(z-\omega)^{2}}-\frac{1}{\omega^{2}}\right).$

It is a meromorphic function in $\mathbb{C}$ with poles of the second order at
every point $\omega\in\Omega$. In half-periods the $\wp$–function takes on
pairwise different values:

$\wp\left(\frac{\omega_{1}}{2}\right)=e_{1},\quad\wp\left(\frac{\omega_{2}}{2}\right)=e_{2},\quad\wp\left(\frac{\omega_{1}+\omega_{2}}{2}\right)=e_{3}.$

We will also need the Weierstrass $\zeta$–function

$\zeta(z)=\zeta(z;\,\omega_{1},\omega_{2})=\frac{1}{z}+\sum_{\omega\,\in\,\Omega,\,\omega\,\neq\,0}\left(\frac{1}{z-\omega}+\frac{1}{\omega}+\frac{z}{\omega^{2}}\right).$

It has the following properties: $\zeta^{\prime}(z)=-\wp(z)$ and
$\zeta(z+\omega_{k})-\zeta(z)=\eta_{k}$, where
$\eta_{k}=2\,\zeta({\omega_{k}}/{2}),\,k=1,2$. There is the Legendre relation
between periods $\omega_{1}$, $\omega_{2}$ and constants $\eta_{1}$,
$\eta_{2}$ (see [2, ch. 3, § 12]):

(1) $\eta_{1}\omega_{2}-\eta_{2}\omega_{1}=2\pi{i}\,.$

Another Weierstrass function is

$\sigma(z)=\sigma(z;\,\omega_{1},\omega_{2})=z\prod_{\omega\,\in\,\Omega,\,\omega\,\neq\,0}\left(1-\frac{z}{\omega}\right)\exp\left\\{{\frac{z}{\omega}+\frac{z^{2}}{2\omega^{2}}}\right\\}.$

It has the following properties:

$\frac{d\log\sigma(z)}{dz}=\zeta(z)\,,\quad\frac{d^{2}\log\sigma(z)}{dz^{2}}=-\wp(z)\,.$
$\sigma(z\pm\omega_{k})=-\exp\left\\{{\pm\,\eta_{k}\left(z\pm\frac{\omega_{k}}{2}\right)}\right\\}\sigma(z)\,,\quad
k=1,2\,.$

We also recall the functions

$\sigma_{1}(z)=\pm\,\frac{\sigma(z\pm{\omega_{1}}/{2})}{\sigma({\omega_{1}}/{2})}\exp\left\\{{\mp\,\frac{\eta_{1}}{2}z}\right\\},$
$\sigma_{2}(z)=\pm\,\frac{\sigma(z\pm{\omega_{2}}/{2})}{\sigma({\omega_{2}}/{2})}\exp\left\\{{\mp\,\frac{\eta_{2}}{2}z}\right\\},$
$\sigma_{3}(z)=\pm\,\frac{\sigma(z\pm{(\omega_{1}+\omega_{2})}/{2})}{\sigma({(\omega_{1}+\omega_{2})}/{2})}\exp\left\\{{\mp\,\frac{\eta_{1}+\eta_{2}}{2}z}\right\\},$

and the Weierstrass invariants

$g_{2}=60\sum_{\omega\,\in\,\Omega,\,\omega\neq 0}\frac{1}{\omega^{4}}\,,\quad
g_{3}=140\sum_{\omega\,\in\,\Omega,\,\omega\,\neq\,0}\frac{1}{\omega^{6}}\,.$

Further we will need explicit expressions for the partial derivative of the
function $\log\sigma(z)=\log\sigma(z;\,\omega_{1},\omega_{2})$ with respect to
the periods $\omega_{1}$ and $\omega_{2}$. In [12] the following fact is
proved.

######  Theorem 2.1.

The partial derivatives of the function
$\log\sigma(z)=\log\sigma(z;\,\omega_{1},\omega_{2})$ with respect to the
periods are equal:

$\frac{\partial\log\sigma(z)}{\partial\omega_{1}}=\frac{1}{2\pi
i}\left[\frac{1}{2}\,\omega_{2}(\wp(z)-\zeta^{2}(z))+\eta_{2}(z\zeta(z)-1)+\frac{g_{2}}{24}\,\omega_{2}z^{2}\right],$
(2) $\frac{\partial\log\sigma(z)}{\partial\omega_{2}}=-\frac{1}{2\pi
i}\left[\frac{1}{2}\,\omega_{1}(\wp(z)-\zeta^{2}(z))+\eta_{1}(z\zeta(z)-1)-\frac{g_{2}}{24}\,\omega_{1}z^{2}\right].$

We also recall the Jacobi theta-functions, which can be expressed via the
Weierstrass elliptic functions (see, e.g. [2, ch. IV, § 19]):

(3)
$\vartheta_{1}(\tau)=\sqrt{\frac{\omega_{1}}{\pi}}\,(e_{1}-e_{2})^{{1}/{4}}(e_{1}-e_{3})^{{1}/{4}}(e_{3}-e_{2})^{{1}/{4}}\exp\left\\{{-\frac{\eta_{1}z^{2}}{2\omega_{1}}}\right\\}\sigma(z)\,,$
$\vartheta_{2}(\tau)=\sqrt{\frac{\omega_{1}}{\pi}}\,(e_{3}-e_{2})^{{1}/{4}}\exp\left\\{{-\frac{\eta_{1}z^{2}}{2\omega_{1}}}\right\\}\sigma_{1}(z)\,,$
$\vartheta_{3}(\tau)=\sqrt{\frac{\omega_{1}}{\pi}}\,(e_{1}-e_{2})^{{1}/{4}}\exp\left\\{{-\frac{\eta_{1}z^{2}}{2\omega_{1}}}\right\\}\sigma_{3}(z)\,,$
$\vartheta_{4}(\tau)=\sqrt{\frac{\omega_{1}}{\pi}}\,(e_{1}-e_{3})^{{1}/{4}}\exp\left\\{{-\frac{\eta_{1}z^{2}}{2\omega_{1}}}\right\\}\sigma_{2}(z)\,;$

here $\tau=z/\omega_{1}$ and $\exp\\{i\pi\omega_{2}/\omega_{1}\\}=q$, $|q|<1$.

We will also need the incomplete elliptic integral of the first kind in the
canonic Legendre form:

(4)
$F(z,k)=\int\limits_{0}^{z}\frac{d\xi}{\sqrt{(1-\xi^{2})(1-k^{2}\xi^{2})}}\,,$

where the parameter $k\in(0,1)$ is called the module of the integral. Putting
$z=1$ in (4), we obtain

(5)
$K(k)=\int\limits_{0}^{1}\frac{d\xi}{\sqrt{(1-\xi^{2})(1-k^{2}\xi^{2})}}\,,$

which is called the complete elliptic integral of the first kind. We will also
denote $K^{\prime}(k)=K(k^{\prime})$ where the parameter
$k^{\prime}=\sqrt{1-k^{2}}$ is called the additional module.

The function $z=\textrm{sn}(w,k)$, inverse to that defined by the integral
(4), is called the Jacobi elliptic sine; its argument can be expressed via
elliptic integrals of the first kind (5). It is elliptic function of the
second kind with the main periods $4K$ and $2iK^{\prime}$.

##  3\. Integral representation

In this section we will deduce an integral representation for the function
mapping conformally an annulus onto a bounded doubly connected polygonal
domain. As we noted above, such integral representations was obtained earlier
by N. I. Akhiezer, G. M. Goluzin and Yu. Komatu. Here we give a slightly
different representation which will be used further and give its short
justification.

Consider a conformal mapping $w=\mathcal{G}(\tau)$ of an annulus
$\mathcal{A}=\\{\tau:q<|\tau|<1\\}$ onto a doubly connected domain
$\mathcal{D}$ in $\mathbb{C}$, the exterior boundary component of which is a
closed $n_{1}$-link polygonal line $\Gamma_{1}$ with vertices at points
$w_{1,i_{1}}$, $1\leq i_{1}\leq n_{1}$, and the interior one is a closed
$n_{2}$-link polygonal line $\Gamma_{2}$ with vertices at points
$w_{2,i_{2}}$, $1\leq i_{2}\leq n_{2}$. Denote by $\alpha_{1,i_{1}}\,\pi$ and
$\alpha_{2,i_{2}}\,\pi$ the interior angles of $\mathcal{D}$ at the vertices
$w_{1,i_{1}}$ and $w_{2,i_{2}}$. The values of angles satisfy

(6)
$\sum\limits_{i_{1}=1}^{n_{1}}\alpha_{1,i_{1}}=n_{1}-2\,,\quad\sum\limits_{i_{2}=1}^{n_{2}}\alpha_{2,i_{2}}=n_{2}+2\,.$

Applying the exponential function $\tau(z)=e^{iz}$, we can consider the
mapping $\mathcal{F}(z)=\mathcal{G}(e^{iz})$ from the horizontal strip
$\\{0<\mathop{\rm Im}\nolimits z<\log q^{-1}\\}$ onto $\mathcal{D}$. It maps
conformally the rectangle $[0,2\pi]\times(0,\log q^{-1})$ with identified
vertical sides onto $\mathcal{D}$. We will seek an integral representation of
this mapping using the Weierstrass elliptic functions with periods
$\omega_{1}=2\pi$ and $\omega_{2}=2i\log q^{-1}$.

It is evident that $\mathcal{F}$ is analytic in the strip $\\{0<\mathop{\rm
Im}\nolimits z<\log q^{-1}\\}$ and is $2\pi$-periodic. Consider the preimages
$z_{1,i_{1}}\in[0,2\pi]$ and $z_{2,i_{2}}=x_{2,i_{2}}+i\log q^{-1}\in[i\log
q^{-1},2\pi+i\log q^{-1}]$ of the vertices of $\mathcal{D}$. Denote by
$\mathcal{Z}$ the set of points which are obtained from the set of preimages
of vertices by all possible shifts to vectors of the lattice $\Omega$,
generated by $\omega_{1}$ and $\omega_{2}$. By the Riemann-Schwarz symmetry
principle, we extend $\mathcal{F}$ to $\mathbb{C}\backslash\mathcal{Z}$ as a
multiply-valued analytic function. It is easy to see that the function
$\displaystyle\frac{\mathcal{F}^{\prime\prime}(z)}{\mathcal{F}^{\prime}(z)}$
is meromorphic and two-periodic in $\mathbb{C}$ with periods $\omega_{1}$ and
$\omega_{2}$. Consider this function in the rectangle
$\Pi_{\omega}=[0,\omega_{1})\times[0,\omega_{2}/i)$, which is its period
parallelogram. We note that the conformal modulus of $\mathcal{D}$ is
$\text{Mod}(\mathcal{D})=\displaystyle\frac{\omega_{2}}{2\omega_{1}i}$. In
$\Pi_{\omega}$ the function
$\displaystyle\frac{\mathcal{F}^{\prime\prime}(z)}{\mathcal{F}^{\prime}(z)}$
has simple poles at the points $z_{1,i_{1}}$ and $z_{2,i_{2}}$ with residues
$\alpha_{1,i_{1}}-1$ and $\alpha_{2,i_{2}}-1$. Therefore, we can restore it by
its singularities with the help of the Weierstrass elliptic function
$\zeta(z)=\zeta(z;\,\omega_{1},\omega_{2})$; it has a unique pole in every
period parallelogram at the point equivalent to $0$ modulo $\Omega$. We have

$\frac{\mathcal{F}^{\prime\prime}(z)}{\mathcal{F}^{\prime}(z)}=c+\sum\limits_{i_{1}=1}^{n_{1}}(\alpha_{1,i_{1}}-1)\,\zeta(z-z_{1,i_{1}})+\sum\limits_{i_{2}=1}^{n_{2}}(\alpha_{2,i_{2}}-1)\,\zeta(z-z_{2,i_{2}})\,,$

where $c$ is a complex constant. Integrating, we obtain

$\log\mathcal{F}^{\prime}(z)=\log
C_{1}+cz+\sum\limits_{i_{1}=1}^{n_{1}}(\alpha_{1,i_{1}}-1)\log\sigma(z-z_{1,i_{1}})+\sum\limits_{i_{2}=1}^{n_{2}}(\alpha_{2,i_{2}}-1)\log\sigma(z-z_{2,i_{2}})\,.$

Then

$\mathcal{F}^{\prime}(z)=C_{1}\exp\\{cz\\}{\prod\limits_{i_{1}=1}^{n_{1}}\sigma^{\alpha_{1,i_{1}}-1}(z-z_{1,i_{1}})}\,{\prod\limits_{i_{2}=1}^{n_{2}}\sigma^{\alpha_{2,i_{2}}-1}(z-z_{2,i_{2}})}\,,$

consequently,

(7)
$\mathcal{F}(z)=C_{1}\int\limits_{0}^{z}\exp\\{c\xi\\}\prod\limits_{i_{1}=1}^{n_{1}}\sigma^{\alpha_{1,i_{1}}-1}(\xi-
z_{1,i_{1}})\prod\limits_{i_{2}=1}^{n_{2}}\sigma^{\alpha_{2,i_{2}}-1}(\xi-
z_{2,i_{2}})\,d\xi+C_{2}\,,$

where $C_{1}\neq 0$ and $C_{2}$ are some complex constants.

Now we will determine the constant $c$ in (7). The function
$\mathcal{F}^{\prime}(z)$ is periodic with period $\omega_{1}$. Taking into
account the equality

$\mathcal{F}^{\prime}(z+\omega_{1})=C_{1}\exp\\{c(z+\omega_{1})\\}\prod\limits_{i_{1}=1}^{n_{1}}\sigma^{\alpha_{1,i_{1}}-1}(z+\omega_{1}-z_{1,i_{1}})\prod\limits_{i_{2}=1}^{n_{2}}\sigma^{\alpha_{2,i_{2}}-1}(z+\omega_{1}-(x_{2,i_{2}}+\omega_{2}/2))\\\
=\exp\\!\Bigg{\\{}\\!c\omega_{1}+\eta_{1}\Bigg{(}\\!\sum\limits_{i_{1}=1}^{n_{1}}(\alpha_{1,i_{1}}-1)(z-z_{1,i_{1}}+\omega_{1}/2)\,\\!+\\!\sum\limits_{i_{2}=1}^{n_{2}}(\alpha_{2,i_{2}}-1)(z-x_{2,i_{2}}+(\omega_{1}-\omega_{2})/2)\\!\Bigg{)}\\!\Bigg{\\}}\mathcal{F}^{\prime}(z),$

we conclude that

$c\omega_{1}+\eta_{1}\Bigg{(}\sum\limits_{i_{1}=1}^{n_{1}}(\alpha_{1,i_{1}}-1)(z-z_{1,i_{1}}+\omega_{1}/2)\,\\\
+\sum\limits_{i_{2}=1}^{n_{1}}(\alpha_{2,i_{2}}-1)(z-x_{2,i_{2}}+(\omega_{1}-\omega_{2})/2)\Bigg{)}\equiv
0\quad(\text{mod}\>2\pi i)\,.$

Using (6), we have

(8)
$c\omega_{1}-\eta_{1}\omega_{2}-\eta_{1}\Bigg{(}\sum\limits_{i_{1}=1}^{n_{1}}(\alpha_{1,i_{1}}-1)\,z_{1,i_{1}}+\sum\limits_{i_{2}=1}^{n_{2}}(\alpha_{2,i_{2}}-1)\,x_{2,i_{2}}\Bigg{)}\equiv
0\quad(\text{mod}\;2\pi i)\,.$

From Legendre’s relation (1) and (8) it follows that

(9)
$c=\frac{\eta_{1}}{\omega_{1}}\Bigg{(}\sum\limits_{i_{1}=1}^{n_{1}}(\alpha_{1,i_{1}}-1)\,z_{1,i_{1}}+\sum\limits_{i_{2}=1}^{n_{2}}(\alpha_{2,i_{2}}-1)\,x_{2,i_{2}}\Bigg{)}+\eta_{2}\,.$

######  Theorem 3.1.

The function $\mathcal{F}$, mapping an annulus
$\mathcal{A}=\\{\tau:\,q<|\tau|<1\\}$ onto the doubly connected polygonal
domain $\mathcal{D}$, has the form (7), where $z=-i\log\tau$. In (7) the
points $z_{1,i_{1}}$ and $z_{2,i_{2}}$ are the preimages of the vertices of
$\mathcal{D}$, the constant $c$ has the form (9), $C_{1}\neq 0$ and $C_{2}$
are some complex constants.

From (3) we can obtain an integral representation of the needed conformal
mapping via Jacobi theta-functions.

######  Corollary 3.1.

The conformal mapping $\mathcal{F}$, described in Theorem 3.1, has the form

(10)
$\mathcal{F}(z)=C^{\prime}_{1}\int\limits_{0}^{z}\exp\left\\{\Big{(}-\eta_{1}\frac{\omega_{2}}{\omega_{1}}+\eta_{2}\Big{)}\xi\right\\}\\\
\times\prod\limits_{i_{1}=1}^{n_{1}}\vartheta_{1}^{\alpha_{1,i_{1}}-1}\left(\frac{\xi-
z_{1,i_{1}}}{\omega_{1}}\right)\,\prod\limits_{i_{2}=1}^{n_{2}}\vartheta_{1}^{\alpha_{2,i_{2}}-1}\left(\frac{\xi-
z_{2,i_{2}}}{\omega_{1}}\right)d\xi+C_{2}\,.$

We note that since theta-functions are expressed by rapidly convergent series,
formula (10) is more convenient for numerical calculations than (7).

Now we will consider an example of conformal mapping onto a doubly connected
polygonal domain and find its accessory parameters. This example will be used
below in Section 5.

######  Example 3.1.

As a doubly connected polygonal region $\mathcal{D}$, we take the rectangle
$\Pi=(-1,1)\times(-b,b)$, $b>0$, with rectilinear slit along the segment
$[a_{1},a_{2}]$ of the real axis, $-1<a_{1}<a_{2}<1$. Now we will describe the
function $\mathcal{F}(z)$, that maps conformally the rectangle
$[-\pi,\pi]\times(0,\log q^{-1})$ with identified vertical sides onto
$\mathcal{D}$.

Since $\mathcal{D}$ is symmetric with respect to the real axis, we may assume
that the preimages of the vertices of $\Pi$ are symmetric with respect to the
imaginary axis, i.e.

$z_{1,1}=\beta_{1,1},\quad z_{1,2}=\beta_{1,2},\quad
z_{1,3}=-\beta_{1,2},\quad z_{1,4}=-\beta_{1,1},\quad
0<\beta_{1,1}<\beta_{1,2}<\pi,$

besides, $z_{2,1}=\pi+i\log q^{-1}$, $z_{2,2}=i\log q^{-1}$. In addition,
$\mathcal{F}(z_{1,i_{1}})=w_{1,i_{1}}$, $1\leq i_{1}\leq 4$, where
$w_{1,1}=1+ib,w_{1,2}=-1+ib,w_{1,3}=-1-ib,w_{1,4}=1-ib$, and
$\mathcal{F}(z_{2,1})=w_{2,1}=a_{1}$, $\mathcal{F}(z_{2,2})=w_{2,2}=a_{2}$.

By Theorem 3.1, the function $\mathcal{F}$ that maps an annulus
$\mathcal{A}=\\{\tau:\;q<|\tau|<1\\}$ onto $\mathcal{D}$, is defined by the
formula

(11)
$\mathcal{F}(z)=C_{1}\int\limits_{0}^{z}\exp\\{c\xi\\}\prod\limits_{i_{1}=1}^{4}\sigma^{-1/2}(\xi-
z_{1,i_{1}})\prod\limits_{i_{2}=1}^{2}\sigma(\xi-z_{2,i_{2}})\,d\xi+C_{2}\,,$

where $z=-i\log\tau$, $c={\eta_{1}}/2+\eta_{2}$,
$\sigma(z)=\sigma(z;\,2\pi,2i\log q^{-1})$.

Now we will find the accessory parameters in (11). Using the Riemann–Schwarz
symmetry principle, we can consider the conformal mapping
$w=\mathcal{G}(\tau)$ of the domain
$\mathcal{A}^{+}=\\{\tau:\;q<|\tau|<1,\;\text{Im}\,\tau>0\\}$ onto
${{\Pi}}^{+}=\\{w:\;-1<\mathop{\rm Re}\nolimits w<1,\;0<\mathop{\rm
Im}\nolimits w<b\\}$ and then extend it up to the conformal mapping of
$\mathcal{A}$ onto $\mathcal{D}$.

Let $k=k(b)\in(0,1)$ be defined by

$k^{1/2}=\frac{\vartheta_{2}(0,\exp\\{-\pi b\\})}{\vartheta_{3}(0,\exp\\{-\pi
b\\})}$

and $c_{1}=K(k)$. Then the Jacobi elliptic function
$u(w)=\text{sn}(c_{1}w,k)$, conformally maps the rectangle ${\Pi}^{+}$ onto
the upper half-plane $\mathbb{H}_{u}^{+}$. In this case, the vertices
$w_{1,1}^{+}=1+ib$, $w_{1,2}^{+}=-1+ib$, $w_{1,3}^{+}=-1$, $w_{1,4}^{+}=1$ of
the rectangle ${\Pi}^{+}$ will be mapped to the points

$u_{1,1}=\frac{1}{k}\,,\quad u_{1,2}=-\frac{1}{k}\,,\quad u_{1,3}=-1,\quad
u_{1,4}=1,$

and the points $a_{1}$ and $a_{2}$ will be mapped to the points
$s_{1}=s_{1}(k)=\text{sn}(c_{1}a_{1},k)$ and
$s_{2}=s_{2}(k)=\text{sn}(c_{1}a_{2},k)$.

Now consider the function $\zeta(\tau)=\displaystyle i\log\tau+(\pi/2+i\log
q^{-1})$, mapping conformally $\mathcal{A}^{+}$ onto the rectangle
${\Pi}_{\mathcal{A}^{+}}=\\{\zeta:\;-\pi/2<\text{Re}\,\zeta<\pi/2,\;0<\text{Im}\,\zeta<\ln
q^{-1}\\}$. It maps the preimages of the vertices of $\Pi^{+}$ under the
action of the needed function to the points

$\zeta_{1,1}=\frac{\pi}{2}-\beta_{1,1}+i\log q^{-1},\
\zeta_{1,2}=\frac{\pi}{2}-\beta_{1,2}+i\log q^{-1},\
\zeta_{1,3}=-\frac{\pi}{2}+i\log q^{-1},\ \zeta_{1,4}=\frac{\pi}{2}+i\log
q^{-1},$

and the points corresponding to the end points of the slits are mapped to the
points $\zeta_{2,1}=-\pi/2$ and $\zeta_{2,2}=\pi/2$.

Then we note that the function $v(\zeta)=\text{sn}(c_{2}\zeta,\ell)$ maps
conformally $\Pi_{\mathcal{A}^{+}}$ onto the upper half-plane
$\mathbb{H}_{v}^{+}$; here the parameters $c_{2}$ and $\ell$ satisfy the
equalities

(12) $\frac{\pi K^{\prime}(\ell)}{2K(\ell)}\,=\log q^{-1},\quad
c_{2}=\frac{2K(\ell)}{\pi}\,.$

This function maps the vertices of the rectangle ${\Pi}_{\mathcal{A}^{+}}$ to
the points $v_{1,3}=-1/\ell,v_{1,4}=1/\ell,v_{2,1}=-1$, $v_{2,2}=1$, and the
preimages of the vertices of $\Pi$ are mapped to
$v_{1,1}=\text{sn}(c_{2}\zeta_{1,1},\ell),v_{1,2}=\text{sn}(c_{2}\zeta_{1,2},\ell)$.
Here the parameter $\ell$ is unknown yet, since it is defined by the value
$\log q^{-1}$, connected with the conformal modulus of $\mathcal{D}$ which is
also still unknown. Our next problem is to determine the dependence
$\ell=\ell(k)$.

First we will construct the Moebius transformation $v=v(u)$ which maps
$v_{1,3}$, $v_{1,4}$, $v_{2,1}$ and $v_{2,2}$ to $u_{1,3}$, $u_{1,4}$,
$s_{1}=s_{1}(k)$ and $s_{2}=s_{2}(k)$. Equating the unharmonic ratios of the
corresponding four points, we obtain

$\frac{-1/\ell(k)-1}{-1/\ell(k)+1}\cdot\frac{1/\ell(k)+1}{1/\ell(k)-1}=\frac{-1-s_{2}(k)}{-1-s_{1}(k)}\cdot\frac{1-s_{1}(k)}{1-s_{2}(k)}\,,$

and from this we find

$\ell(k)=\frac{s_{1}(k)s_{2}(k)-1+\sqrt{(s_{1}^{2}(k)-1)(s_{2}^{2}(k)-1)}}{s_{1}(k)-s_{2}(k)}\,.$

Applying the Moebius transformation $v=v(u)$, we obtain that
$v_{{1,1}}=v(u_{{1,1}})$ has the form

$v_{{1,1}}=\frac{\left(k(1+\ell)+(1-\ell)\right)s_{1}+\left(k(1-\ell)+(1+\ell)\right)s_{2}-2ks_{1}s_{2}-2}{\left(k(\ell+1)+(\ell-1)\right)s_{1}+\left(k(\ell-1)+(\ell+1)\right)s_{2}-2\ell
ks_{1}s_{2}-2\ell}\,.$

Using (12) and the fact that the function, inverse to the Jacobi elliptic
sine, is expressed via the incomplete elliptic integral, we have

$c_{2}z_{1,1}=K(\ell)+iK^{\prime}(\ell)-\int\limits_{1/\ell}^{v_{{1,1}}}\frac{d\xi}{\sqrt{(1-\xi^{2})(1-\ell^{2}\xi^{2})}}\,.$

On the other side, using the expressions of $c_{2}$ and $z_{1,1}$ in terms of
elliptic functions, we obtain

$c_{2}z_{1,1}=\frac{2K(\ell)}{\pi}\left(\frac{\pi}{2}-\beta_{1,1}+i\frac{\pi}{2}\frac{K^{\prime}(\ell)}{K(\ell)}\right),$

and from this we find that

$\beta_{1,1}(\ell)=\frac{\pi}{2K(\ell)}\int\limits_{1/\ell}^{v_{{1,1}}(\ell)}\frac{d\xi}{\sqrt{(1-\xi^{2})(1-\ell^{2}\xi^{2})}}\,.$

Similarly we deduce:

$\beta_{1,2}(\ell)=\pi-\frac{\pi}{2K(\ell)}\int\limits_{v_{{1,2}}}^{-1/\ell}\frac{d\xi}{\sqrt{(1-\xi^{2})(1-\ell^{2}\xi^{2})}}\,,$

where

$v_{{1,2}}=\frac{\left(k(1+\ell)-(1-\ell)\right)s_{1}+\left(k(1-\ell)-(1+\ell)\right)s_{2}-2ks_{1}s_{2}+2}{\left(k(\ell+1)-(\ell-1)\right)s_{1}+\left(k(\ell-1)-(\ell+1)\right)s_{2}-2\ell
ks_{1}s_{2}+2\ell}\,.$

From (12) it follows that the conformal modulus of $\mathcal{D}$ is defined by
the formula

$\text{Mod}(\mathcal{D})=\frac{K^{\prime}(\ell)}{4K(\ell)}\,.$

In Table 3.1 we give values of accessory parameters and conformal modulus for
some $a_{1}$ and $a_{2}$, if $b=0.5$.

Table 3.1. Some values of the accessory parameters and the conformal modulus of the rectangle $(-1,1)\times(-0.5,0.5)$, with a linear cut along the segment $[a_{1},a_{2}]$. $a_{1}$ | $a_{2}$ | $\omega_{2}$ | $z_{1,1}$ | $z_{1,2}$ | $\text{Mod}(\mathcal{D})$
---|---|---|---|---|---
$-0.75$ | $0.75$ | $1.5034301914585424\,i$ | $0.4710482171084344$ | $2.670544436481146$ | $0.11963917328211081$
$-0.25$ | $0.25$ | $3.3301140313814135\,i$ | $0.19928075807406562$ | $2.9423118955157292$ | $0.26500205457701553$
$-0.1$ | $0.5$ | $2.979450683132465\,i$ | $0.39772860081051964$ | $3.0299504808966304$ | $0.2370971519595281$
$0.1$ | $0.9$ | $2.1183213413271487\,i$ | $1.2511247023650072$ | $3.0991773846339017$ | $0.16857065626463488$
$-0.5$ | $0.5$ | $2.1382753178673837\,i$ | $0.28701506868055354$ | $2.8545775849092414$ | $0.17015854326499394$

Example 3.1 shows that for simple symmetric domains we can find the accessory
parameters with the help of elliptic functions. For more complicated domains,
we need to apply approximate methods.

##  4\. Families of conformal mappings

Now we consider one parameter family of doubly connected domains
$\mathcal{D}(t)$, obtained from a fixed doubly connected polygonal domain
$\mathcal{D}$ (Fig. 4.1) by making a few disjoint rectilinear slits. We will
assume that the end points of the slits are smooth functions of the parameter
$t$ changing on some segment. For the objects and characteristics, connected
with $\mathcal{D}$, we will use the same notation as in Section 3.

Let us have $m_{1}$ slits going from pairwise different points of $\Gamma_{1}$
at angles $\varphi_{1,j_{1}}\pi$, $1\leq j_{1}\leq m_{1}$, and $m_{2}$ slits
from $\Gamma_{2}$ at angles $\varphi_{2,j_{2}}\pi$, $1\leq j_{2}\leq m_{2}$
(Fig. 4.1). For simplicity, we will assume that all the slits proceed from
non-angular points of the boundary. The general case can be considered
similarly (see Remark 4.2 below). Denote by
${z}_{1,j_{1},0}(t)\in[0,\omega_{1}]$ the preimages of the end points of slits
proceeding from $\Gamma_{1}$, and by
${z}_{2,j_{2},0}(t)={x}_{2,j_{2},0}(t)+\omega_{2}(t)/2\in[\omega_{2}(t)/2,\omega_{1}+\omega_{2}(t)/2]$
the preimages of the end points of slits proceeding from $\Gamma_{2}$. We will
also denote by $z_{1,j_{1},1}(t)$ and $z_{1,j_{1},2}(t)$ new angular points of
the exterior boundary component with inner angles $\varphi_{1,j_{1}}\pi$ and
$(1-\varphi_{1,j_{1}})\pi$, and by $z_{2,j_{2},1}(t)$ and $z_{2,j_{2},2}(t)$
we denote new angular points of the interior boundary component with inner
angles $\varphi_{2,j_{2}}\pi$ and $(1-\varphi_{2,j_{2}})\pi$. Now we will
describe smooth one parameter families of functions $\mathcal{F}(z,t)$, which
conformally map the annuli $\mathcal{A}(t)=\\{\tau:\;q(t)<|\tau|<1\\}$ onto
$\mathcal{D}(t)$; here $q(t)=\exp\\{-2\pi m(t)\\}$, $m(t)$ is the conformal
modulus of $\mathcal{D}(t)$.

Figure 4.1. Doubly connected polygonal domain $\mathcal{D}$ and domain
$\mathcal{D}(t)$ obtained from $\mathcal{D}$ by drawing a finite number of
slits.

Here, in contrast to the classical Loewner–Komatu method, we consider the case
of several slits and do not assume that the family $\mathcal{D}(t)$ is
monotonous with respect to the parameter $t$.

Now we will find a PDE for the family $\mathcal{F}(z,t)$ and then we will
deduce from it a system of ODE to determine the accessory parameters, in
particular, the conformal modulus $\text{Mod}(\mathcal{D}(t))$ for all values
of parameter $t$.

So we consider a smooth one parameter family of conformal mappings

(13)
$\mathcal{F}(z,t)=C_{1}(t)\int\limits_{0}^{z}\exp\\{c(t)\,\xi\\}\prod\limits_{i_{1}=1}^{n_{1}}\sigma^{\alpha_{1,i_{1}}-1}(\xi-
z_{1,i_{1}}(t))\prod\limits_{i_{2}=1}^{n_{2}}\sigma^{\alpha_{2,i_{2}}-1}(\xi-
z_{2,i_{2}}(t))\,\\\
\times\prod\limits_{j_{1}=1}^{m_{1}}s_{1,j_{1}}(\xi,t)\prod\limits_{j_{2}=1}^{m_{2}}s_{2,j_{2}}(\xi,t)\,d\xi+C_{2}\,,$

where

$s_{1,j_{1}}(z,t)=\sigma(z-{z}_{1,j_{1},0}(t))\,\sigma^{\varphi_{1,j_{1}}-1}(z-z_{1,j_{1},1}(t))\,\sigma^{-\varphi_{1,j_{1}}}(z-z_{1,j_{1},2}(t))\,,$
$s_{2,j_{2}}(z,t)=\sigma(z-{z}_{2,j_{2},0}(t))\,\sigma^{\varphi_{2,j_{2}}-1}(z-z_{2,j_{2},1}(t))\,\sigma^{-\varphi_{2,j_{2}}}(z-z_{2,j_{2},2}(t))\,,$
$c(t)=\frac{\eta_{1}(t)}{\omega_{1}}\Bigg{[}\sum\limits_{i_{1}=1}^{n_{1}}(\alpha_{1,i_{1}}-1)\,z_{1,i_{1}}(t)+\sum\limits_{i_{2}=1}^{n_{2}}(\alpha_{2,i_{2}}-1)\,x_{2,i_{2}}(t)\,\\\
+\sum\limits_{j_{1}=1}^{m_{1}}\Big{(}{z}_{1,j_{1},0}(t)+(\varphi_{1,j_{1}}-1)\,z_{1,j_{1},1}(t)-\varphi_{1,j_{1}}\,z_{1,j_{1},2}(t)\Big{)}\\\
+\sum\limits_{j_{2}=1}^{m_{2}}\Big{(}{x}_{2,j_{2},0}(t)+(\varphi_{2,j_{2}}-1)\,x_{2,j_{2},1}(t)-\varphi_{2,j_{2}}\,x_{2,j_{2},2}(t)\Big{)}\Bigg{]}+\eta_{2}(t)\,.$

Here $\sigma(z)=\sigma(z;\,\omega_{1},\omega_{2}(t))$,
$\,\eta_{1}(t)=2\zeta({\omega_{1}}/{2};\,\omega_{1},\omega_{2}(t))\,$ and
$\,\eta_{2}(t)=2\zeta({\omega_{2}(t)}/{2};\,\omega_{1},$ $\omega_{2}(t))$. The
conformal modulus of $\mathcal{D}(t)$ equals

$\text{Mod}(\mathcal{D}(t))=\displaystyle\frac{\omega_{2}(t)}{2\omega_{1}i}\,.$

For a fixed $t$, the function $\mathcal{F}(z,t)$ is periodic with period
$\omega_{1}=2\pi$. By the symmetry principle, we extend $\mathcal{F}(z,t)$
through the segments with endpoints which are the preimages of the vertices of
the boundary components. It is evident that the extension satisfies the
equalities

$\mathcal{F}(z+\omega_{1},t)=\mathcal{F}(z,t)\,,\quad\mathcal{F}(z+\omega_{2}(t),t)=e^{i\alpha}\mathcal{F}(z,t)+\beta\,.$

Differentiating with respect to $t$ and $z$, we obtain

$\dot{\mathcal{F}}(z+\omega_{1},t)=\dot{\mathcal{F}}(z,t),\,\quad\dot{\omega}_{2}(t)\,\mathcal{F}^{\prime}(z+\omega_{2}(t),t)+\dot{\mathcal{F}}(z+\omega_{2}(t),t)=e^{i\alpha}\dot{\mathcal{F}}(z,t)\,,$
$\mathcal{F}^{\prime}(z+\omega_{1},t)=\mathcal{F}^{\prime}(z,t)\,,\quad\mathcal{F}^{\prime}(z+\omega_{2}(t),t)=e^{i\alpha}\mathcal{F}^{\prime}(z,t)\,.$

Here and further the dot above a letter denotes the differentiation with
respect to $t$, and the prime denotes differentiation with respect to $z$.
Therefore, we have

$\frac{\dot{\mathcal{F}}\left(z+\omega_{1},t\right)}{\mathcal{F}^{\prime}(z+\omega_{1},t)}=\frac{\dot{\mathcal{F}}(z,t)}{\mathcal{F}^{\prime}(z,t)}\,,$
$\frac{\dot{\mathcal{F}}\left(z+\omega_{2}(t),t\right)}{\mathcal{F}^{\prime}(z+\omega_{2}(t),t)}+\dot{\omega}_{2}(t)=\frac{\dot{\mathcal{F}}(z,t)}{\mathcal{F}^{\prime}(z,t)}\,.$

Consequently, the function

$\mathcal{H}(z,t):=\frac{\dot{\mathcal{F}}(z,t)}{\mathcal{F}^{\prime}(z,t)}$

satisfies

(14)
$\mathcal{H}\left(z+\omega_{1},t\right)-\mathcal{H}(z,t)=0\,,\quad\mathcal{H}\left(z+\omega_{2}(t),t\right)-\mathcal{H}(z,t)=-\dot{\omega}_{2}(t)\,.$

Now we will write the Taylor expansion of $\mathcal{F}(z,t)$ near the point
$z_{1,k_{1}}(t)$:

$\mathcal{F}(z,t)=A_{0}+A_{1}(t)\left(z-z_{1,k_{1}}(t)\right)^{\alpha_{1,k_{1}}}+\ldots\,,\quad
1\leq k_{1}\leq n_{1}\,;$

here $A_{0}$ does not depend on $t$. Then

$\mathcal{F}^{\prime}(z,t)=\alpha_{1,k_{1}}\,A_{1}(t)\left(z-z_{1,k_{1}}(t)\right)^{\alpha_{1,k_{1}}-1}+\ldots\,,$
$\dot{\mathcal{F}}(z,t)=-\alpha_{1,k_{1}}\,A_{1}(t)\left(z-z_{1,k_{1}}(t)\right)^{\alpha_{1,k_{1}}-1}\dot{z}_{1,k_{1}}(t)+\ldots\,$

and, therefore,

(15) $\dot{z}_{1,k_{1}}(t)=-\mathcal{H}(z_{1,k_{1}}(t),t)\,,\quad 1\leq
k_{1}\leq n_{1}\,.$

Similarly, we obtain

(16) $\dot{z}_{2,k_{2}}(t)=-\mathcal{H}(z_{2,k_{2}}(t),t)\,,\quad 1\leq
k_{2}\leq n_{2}\,,$ (17)
$\dot{z}_{1,\ell_{1},j}(t)=-\mathcal{H}(z_{1,\ell_{1},j}(t),t)\,,\quad
1\leq\ell_{1}\leq m_{1}\,,\quad j=1,2\,,$ (18)
$\dot{z}_{2,\ell_{2},j}(t)=-\mathcal{H}(z_{2,\ell_{2},j}(t),t)\,,\quad
1\leq\ell_{2}\leq m_{2}\,,\quad j=1,2\,.$

Now we will write the Taylor expansion for $\mathcal{F}(z,t)$ near the points
${z}_{1,\ell_{1},0}(t)$:

(19)
$\mathcal{F}(z,t)=\mathcal{E}_{1,\ell_{1}}(t)+\frac{1}{2}\,{\mathcal{F}^{\prime\prime}\left({z}_{1,\ell_{1},0}(t),t\right)}\left(z-{z}_{1,\ell_{1},0}(t)\right)^{2}+\ldots\,,\quad
1\leq\ell_{1}\leq m_{1}\,,$

where $\mathcal{E}_{1,\ell_{1}}(t)=\mathcal{F}(z_{1,\ell_{1},0}(t),t)$ is the
end point of the $\ell_{1}$th slit, entering from $\Gamma_{1}$. From the
expansion it follows that

$\mathcal{F}^{\prime}(z,t)=\mathcal{F}^{\prime\prime}\left({z}_{1,\ell_{1},0}(t),t\right)\left(z-{z}_{1,\ell_{1},0}(t)\right)+\ldots\,,$
$\dot{\mathcal{F}}(z,t)=\dot{\mathcal{E}}_{1,\ell_{1}}(t)-\mathcal{F}^{\prime\prime}\left({z}_{1,\ell_{1},0}(t),t\right)\left(z-{z}_{1,\ell_{1},0}(t)\right)\dot{{z}}_{1,\ell_{1},0}(t)+\ldots\,,$

and, thus,

$\mathcal{H}(z,t)=\frac{\mathcal{L}_{1,\ell_{1}}(t)}{z-{z}_{1,\ell_{1},0}(t)}+\text{O}(1)\,,\quad
z\rightarrow{z}_{1,\ell_{1},0}(t)\,,$

where

$\mathcal{L}_{1,\ell_{1}}(t):=\frac{\dot{\mathcal{E}}_{1,\ell_{1}}(t)}{\mathcal{F}^{\prime\prime}\left({z}_{1,\ell_{1},0}(t),t\right)},\quad
1\leq\ell_{1}\leq m_{1}\,.$

Similarly,

$\mathcal{L}_{2,\ell_{2}}(t):=\frac{\dot{\mathcal{E}}_{2,\ell_{2}}(t)}{\mathcal{F}^{\prime\prime}\left({z}_{2,\ell_{2},0}(t),t\right)},\quad
1\leq\ell_{2}\leq m_{2}\,.$

Now we will find
$\mathcal{F}^{\prime\prime}\left({z}_{1,\ell_{1},0}(t),t\right)$. We have

$\mathcal{F}^{\prime\prime}(z,t)=C_{1}(t)\exp\\{c(t)\,z\\}\prod\limits_{i_{1}=1}^{n_{1}}\sigma^{\alpha_{1,i_{1}}-1}(z-z_{1,i_{1}}(t))\prod\limits_{i_{2}=1}^{n_{2}}\sigma^{\alpha_{2,i_{2}}-1}(z-z_{2,i_{2}}(t))\,\\\
\times\prod\limits_{j_{1}=1}^{m_{1}}s_{2,j_{1}}(z,t)\prod\limits_{j_{2}=1}^{m_{2}}s_{2,j_{2}}(z,t)\Bigg{[}\mathcal{Q}(z,t)+\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{Q}_{1,i_{1}}(z,t)+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{Q}_{2,i_{2}}(z,t)\Bigg{]},$

where

$\mathcal{Q}(z,t)=c(t)+\sum\limits_{i_{1}=1}^{n_{1}}(\alpha_{1,i_{1}}-1)\,\zeta(z-z_{1,i_{1}}(t))+\sum\limits_{i_{2}=1}^{n_{2}}(\alpha_{2,i_{2}}-1)\,\zeta(z-z_{2,i_{2}}(t))\,,$
$\mathcal{Q}_{1,i_{1}}(z,t)=\zeta(z-{z}_{1,j_{1},0}(t))+(\varphi_{1,j_{1}}-1)\,\zeta(z-z_{1,j_{1},1}(t))-\varphi_{1,j_{1}}\,\zeta(z-z_{1,j_{1},2}(t))\,,$
$\mathcal{Q}_{2,i_{2}}(z,t)=\zeta(z-{z}_{2,j_{2},0}(t))+(\varphi_{2,j_{2}}-1)\,\zeta(z-z_{2,j_{2},1}(t))-\varphi_{2,j_{2}}\,\zeta(z-z_{2,j_{2},2}(t))\,.$

Consequently, as $z\rightarrow{z}_{1,\ell_{1},0}(t)$, we obtain

(20)
$\mathcal{F}^{\prime\prime}\left({z}_{1,\ell_{1},0}(t),t\right)=C_{1}(t)\exp\\{c(t)\,{z}_{1,\ell_{1},0}(t)\\}\,\\\
\times\prod\limits_{i_{1}=1}^{n_{1}}\sigma^{\alpha_{1,i_{1}}-1}({z}_{1,\ell_{1},0}(t)-z_{1,i_{1}}(t))\,\prod\limits_{i_{2}=1}^{n_{2}}\sigma^{\alpha_{2,i_{2}}-1}({z}_{1,\ell_{1},0}(t)-z_{2,i_{2}}(t))\,\\\
\times\prod\limits_{j_{1}=1}^{m_{1}}\sigma^{\varphi_{1,j_{1}}-1}({z}_{1,\ell_{1},0}(t)-z_{1,j_{1},1}(t))\,\sigma^{-\varphi_{1,j_{1}}}({z}_{1,\ell_{1},0}(t)-z_{1,j_{1},2}(t))\,\prod\limits_{j_{2}=1}^{m_{2}}s_{2,j_{2}}({z}_{1,\ell_{1},0}(t),t)\,.$

Similarly,

(21)
$\mathcal{F}^{\prime\prime}\left({z}_{2,\ell_{2},0}(t),t\right)=C_{1}(t)\exp\\{c(t)\,{z}_{2,\ell_{2},0}(t)\\}\,\\\
\times\prod\limits_{i_{1}=1}^{n_{1}}\sigma^{\alpha_{1,i_{1}}-1}({z}_{2,\ell_{2},0}(t)-z_{1,i_{1}}(t))\,\prod\limits_{i_{2}=1}^{n_{2}}\sigma^{\alpha_{2,i_{2}}-1}({z}_{2,\ell_{2},0}(t)-z_{2,i_{2}}(t))\,\\\
\times\prod\limits_{j_{1}=1}^{m_{1}}s_{1,j_{1}}({z}_{2,\ell_{2},0}(t),t)\prod\limits_{j_{2}=1}^{m_{2}}\sigma^{\varphi_{2,j_{2}}-1}({z}_{2,\ell_{2},0}(t)-z_{2,j_{2},1}(t))\,\sigma^{-\varphi_{2,j_{2}}}({z}_{2,\ell_{2},0}(t)-z_{2,j_{2},2}(t))\,.$

The function

$\mathcal{S}(z,t):=\mathcal{H}(z,t)-\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)\,\zeta(z-{z}_{1,j_{1},0}(t))-\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\,\zeta(z-{z}_{2,j_{2},0}(t))$

has removable singularities only at the points which are the preimages of the
end points of the slits and at points equivalent to them modulo the lattice.
Consequently, it has a holomorphic extension to the whole plane $\mathbb{C}$.
From (14) we have

$\mathcal{S}(z+\omega_{1},t)-\mathcal{S}(z,t)=-\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)\,\zeta(z+\omega_{1}-{z}_{1,j_{1},0}(t))-\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\,\zeta(z+\omega_{1}-{z}_{2,j_{2},0}(t))\,$
$+\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)\,\zeta(z-{z}_{1,j_{1},0}(t))+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\,\zeta(z-{z}_{2,j_{2},0}(t))$
(22)
$=-\eta_{1}(t)\Bigg{(}\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\Bigg{)},$
(23)
$\mathcal{S}(z+\omega_{2}(t),t)-\mathcal{S}(z,t)=-\dot{\omega}_{2}(t)-\eta_{2}(t)\Bigg{(}\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\Bigg{)}.$

Because of (22) and (23), the function $\mathcal{S}$ grows at infinity no
faster that a linear one, therefore, $\mathcal{S}(z,t)=\alpha(t)z+\beta(t)$.
Thus,

(24)
$\mathcal{H}(z,t)=\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)\,\zeta(z-{z}_{1,j_{1},0}(t))+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\,\zeta(z-{z}_{2,j_{2},0}(t))+\alpha(t)\,z+\beta(t)\,.$

From (22) for $z=0$ it follows that

(25)
$\alpha(t)=-\frac{\eta_{1}(t)}{\omega_{1}}\Bigg{(}\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\Bigg{)}.$

Now we will demand that $\mathcal{F}(0,t)$ does not depend on $t$. It is
possible since the conformal mappings of concentric annuli onto domains are
defined up to rotation of the annuli. So we have $\mathcal{F}(0,t)=0$ and
$\mathcal{H}(0,t)=0$. Then

(26)
$\beta(t)=\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)\,\zeta({z}_{1,j_{1},0}(t))+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\,\zeta({z}_{2,j_{2},0}(t)).$

From (23) for $z=0$ it follows that

$\dot{\omega}_{2}(t)=-\alpha(t)\,\omega_{2}(t)-\eta_{2}(t)\Bigg{(}\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\Bigg{)}\\\
=\frac{\eta_{1}(t)\,\omega_{2}(t)-\eta_{2}(t)\,\omega_{1}}{\omega_{1}}\Bigg{(}\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\Bigg{)},$

and, with the help of Legendre’s relation (1), we obtain

(27)
$\dot{\omega}_{2}(t)=i\Bigg{(}\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\Bigg{)}.$

At last, from (24), (25) and (26) we deduce that

(28)
$\mathcal{H}(z,t)=\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)\,\mathcal{K}_{1,j_{1}}(z,t)\,+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\,\mathcal{K}_{2,j_{2}}(z,t)\,,$

where

$\mathcal{K}_{1,j_{1}}(z,t)=\zeta(z-{z}_{1,j_{1},0}(t))-\frac{\eta_{1}(t)}{\omega_{1}}\,z+\zeta({z}_{1,j_{1},0}(t))\,,$
$\mathcal{K}_{2,j_{2}}(z,t)=\zeta(z-{z}_{2,j_{2},0}(t))-\frac{\eta_{1}(t)}{\omega_{1}}\,z+\zeta({z}_{2,j_{2},0}(t))\,.$

We note that the functions $\mathcal{K}_{1,j_{1}}$ and $\mathcal{K}_{2,j_{2}}$
are similar to the Villat kernel entering into the integral representation
giving a solution to the Schwarz problem about recovering an analytic function
by the boundary values of its real part (see, e.g. [2, ch. 10, §. 56]).
Therefore, we proved the following fact.

######  Theorem 4.1.

The smooth family of conformal mappings $\mathcal{F}(z,t)$, defined above,
satisfies the PDE

(29)
$\frac{\dot{\mathcal{F}}(z,t)}{\mathcal{F}^{\prime}(z,t)}=\mathcal{H}(z,t)\,.$

Here $\mathcal{H}(z,t)$ is defined by (28), and the Weierstrass
$\zeta$-function from (28) has periods $\omega_{1}=2\pi$ and
$\omega_{2}=\omega_{2}(t)$ satisfying (27).

Now we will find a differential equation for ${z}_{1,j_{1},0}(t)$. To do this,
we will write $\dot{\mathcal{F}}^{\prime}({z}_{1,j_{1},0}(t),t)$ in two
different ways. From (19) it follows that

$\mathcal{F}^{\prime}(z,t)=\mathcal{F}^{\prime\prime}({z}_{1,j_{1},0}(t),t)\,(z-{z}_{1,j_{1},0}(t))+\dots\,.$

Then

(30)
$\dot{{z}}_{1,j_{1},0}(t)=-\frac{\dot{\mathcal{F}}^{\prime}({z}_{1,j_{1},0}(t),t)}{\mathcal{F}^{\prime\prime}({z}_{1,j_{1},0}(t),t)}\,.$

On the other side, as $z\rightarrow{z}_{1,\ell_{1},0}(t)$, by Theorem 4.1, we
have

(31)
$\dot{\mathcal{F}}^{\prime}({z}_{1,\ell_{1},0}(t),t)=\mathcal{F}^{\prime\prime}({z}_{1,\ell_{1},0}(t),t)\,\mathcal{H}({z}_{1,\ell_{1},0}(t),t)+\mathcal{F}^{\prime}({z}_{1,\ell_{1},0}(t),t)\,\mathcal{H}^{\prime}({z}_{1,\ell_{1},0}(t),t)\,.$

Comparing (30) and (31), and taking into account (20), we obtain

(32)
$\dot{{z}}_{1,\ell_{1},0}(t)=-\sum\limits_{j_{1}=1,\,j_{1}\neq\ell_{1}}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)\,\mathcal{K}_{1,j_{1}}({{z}}_{1,\ell_{1},0}(t),t)\,-\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\,\mathcal{K}_{2,j_{2}}({{z}}_{1,\ell_{1},0}(t),t)\,\\\
-\,\mathcal{L}_{1,\ell_{1}}(t)\Bigg{[}\mathcal{Q}({z}_{1,\ell_{1},0}(t),t)\,+\sum\limits_{j_{1}=1,\,j_{1}\neq\ell_{1}}^{m_{1}}\mathcal{Q}_{1,i_{1}}({z}_{1,\ell_{1},0}(t),t)\,+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{Q}_{2,i_{2}}({z}_{1,\ell_{1},0}(t),t)\Bigg{]}.$

Similarly,

(33)
$\dot{{z}}_{2,\ell_{2},0}(t)=-\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)\,\mathcal{K}_{1,j_{1}}({{z}}_{2,\ell_{2},0}(t),t)\,-\sum\limits_{j_{2}=1,\,j_{2}\neq\ell_{2}}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\,\mathcal{K}_{2,j_{2}}({{z}}_{2,\ell_{2},0}(t),t)\,\\\
-\,\mathcal{L}_{2,\ell_{2}}(t)\Bigg{[}\mathcal{Q}({z}_{2,\ell_{2},0}(t),t)\,+\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{Q}_{1,i_{1}}({z}_{2,\ell_{2},0}(t),t)\,+\sum\limits_{j_{2}=1,\,j_{2}\neq\ell_{2}}^{m_{2}}\mathcal{Q}_{2,i_{2}}({z}_{2,\ell_{2},0}(t),t)\Bigg{]}.$

Now we will find a differential equation for $C_{1}(t)$. On one side, from
Theorem 4.1 it follows that

$\frac{\dot{\mathcal{F}}^{\prime}(0,t)}{{\mathcal{F}}^{\prime}(0,t)}=-\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)\,\mathcal{P}({z}_{1,j_{1},0}(t),t)-\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\,\mathcal{P}({z}_{2,j_{2},0}(t),t)\,,$

where
$\mathcal{P}(z,t)=\wp(z;\,\omega_{1},\omega_{2}(t))+{\eta_{1}(t)}/{\omega_{1}}$.

On the other side, from (13) we have

$\frac{\dot{\mathcal{F}}^{\prime}(0,t)}{{{\mathcal{F}}^{\prime}(0,t)}}=\frac{\dot{C}_{1}(t)}{C_{1}(t)}\,+\sum\limits_{i_{1}=1}^{n_{1}}(\alpha_{1,i_{1}}-1)\,\mathcal{R}(z_{1,i_{1}}(t),t)\,+\sum\limits_{i_{2}=1}^{n_{2}}(\alpha_{2,i_{2}}-1)\,\mathcal{R}(z_{2,i_{2}}(t),t)\,\\\
+\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{R}({z}_{1,j_{1},0}(t),t)+(\varphi_{1,j_{1}}-1)\,\mathcal{R}(z_{1,j_{1},1}(t),t)-\varphi_{1,j_{1}}\,\mathcal{R}(z_{1,j_{1},2}(t),t)\,\\\
+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{R}({z}_{2,j_{2},0}(t),t)+(\varphi_{2,j_{2}}-1)\,\mathcal{R}(z_{2,j_{2},1}(t),t)-\varphi_{2,j_{2}}\,\mathcal{R}(z_{2,j_{2},2}(t),t)\,,$

where

$\mathcal{R}(z(t),t)=\zeta(z(t);\,\omega_{1},\omega_{2}(t))\,\dot{z}(t)+\dot{\omega}_{2}(t)\,\frac{\partial\log\sigma(z(t);\,\omega_{1},\omega_{2}(t))}{\partial\omega_{2}}\,.$

Then

(34)
$-\frac{\dot{C}_{1}(t)}{C_{1}(t)}=\sum\limits_{i_{1}=1}^{n_{1}}(\alpha_{1,i_{1}}-1)\,\mathcal{R}(z_{1,i_{1}}(t),t)\,+\sum\limits_{i_{2}=1}^{n_{2}}(\alpha_{2,i_{2}}-1)\,\mathcal{R}(z_{2,i_{2}}(t),t)\,\\\
+\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{R}({z}_{1,j_{1},0}(t),t)+(\varphi_{1,j_{1}}-1)\,\mathcal{R}(z_{1,j_{1},1}(t),t)-\varphi_{1,j_{1}}\,\mathcal{R}(z_{1,j_{1},2}(t),t)\,\\\
+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{R}({z}_{2,j_{2},0}(t),t)+(\varphi_{2,j_{2}}-1)\,\mathcal{R}(z_{2,j_{2},1}(t),t)-\varphi_{2,j_{2}}\,\mathcal{R}(z_{2,j_{2},2}(t),t)\,\\\
+\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)\,\mathcal{P}({z}_{1,j_{1},0}(t),t)\,+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\,\mathcal{P}({z}_{2,j_{2},0}(t),t)\,.$

Thus, we have the following statement.

######  Theorem 4.2.

The accessory parameters of the family $\mathcal{F}(z,t)$ satisfy the system
of ODEs (15) – (18), (27), (32), (33) and (34) where the function
$\mathcal{H}(z,t)$ is defined by (28). In (34) the partial derivative of
$\log\sigma(z)$ with respects to the period $\omega_{2}$ is given by the
formula (2).

######  Corollary 4.1.

The conformal modulus $m(t)=\textup{Mod}(\mathcal{D}(t))$ satisfies the
differential equation

$\dot{m}(t)=\frac{1}{2\omega_{1}}\Bigg{(}\sum\limits_{j_{1}=1}^{m_{1}}\mathcal{L}_{1,j_{1}}(t)+\sum\limits_{j_{2}=1}^{m_{2}}\mathcal{L}_{2,j_{2}}(t)\Bigg{)}.$

######  Remark 4.1.

We note that the right-hand sides of (15) – (18), (27), (32), (33) and (34)
have singularities at the initial moment (say, $t=0$). This is caused by the
fact that the lengths of the slits tend to zero as $t\to 0$, consequently, for
every fixed $1\leq k\leq 2$ and $1\leq j_{k}\leq m_{k}$ the points
$z_{k,j_{k},0}(t)$, $z_{k,j_{k},1}(t)$ and $z_{k,j_{k},2}(t)$ have the same
limit. To avoid degeneration when solving the Cauchy problem for the system,
in practice we change initial values of the parameters by the values of order
$10^{-12}$.

######  Remark 4.2.

Consider the case when we make a new $(m_{1}+1)$th slit from the $j$th vertex
of $\Gamma_{1}$ (the case of of a slit going from a vertex of $\Gamma_{2}$ is
studied simiklarly). Then the family of conformal mappings (13) will be
rewritten in the form

$\mathcal{F}(z,t)=C_{1}(t)\int\limits_{0}^{z}\exp\\{c(t)\,\xi\\}\prod\limits_{i_{1}=1,\,i_{1}\neq
j}^{n_{1}}\sigma^{\alpha_{1,i_{1}}-1}(\xi-
z_{1,i_{1}}(t))\prod\limits_{i_{2}=1}^{n_{2}}\sigma^{\alpha_{2,i_{2}}-1}(\xi-
z_{2,i_{2}}(t))\,\times\\\
\times\prod\limits_{j_{1}=1}^{m_{1}+1}s_{1,j_{1}}(\xi,t)\prod\limits_{j_{2}=1}^{m_{2}}s_{2,j_{2}}(\xi,t)\,d\xi+C_{2}\,,$

where

$s_{1,j_{1}}(z,t)=\sigma(z-{z}_{1,j_{1},0}(t))\,\sigma^{\varphi_{1,j_{1}}-1}(z-z_{1,j_{1},1}(t))\,\sigma^{-\varphi_{1,j_{1}}}(z-z_{1,j_{1},2}(t))\,,\quad
1\leq j_{1}\leq m_{1},$
$s_{1,m_{1}+1}(z,t)=\sigma(z-{z}_{1,m_{1}+1,0}(t))\,\sigma^{\varphi_{1,m_{1}+1,1}-1}(z-z_{1,m_{1}+1,1}(t))\,\sigma^{\varphi_{1,m_{1}+1,2}-1}(z-z_{1,m_{1}+1,2}(t))\,,$
$s_{2,j_{2}}(z,t)=\sigma(z-{z}_{2,j_{2},0}(t))\,\sigma^{\varphi_{2,j_{2}}-1}(z-z_{2,j_{2},1}(t))\,\sigma^{-\varphi_{2,j_{2}}}(z-z_{2,j_{2},2}(t))\,,$
$c(t)=\frac{\eta_{1}(t)}{\omega_{1}}\Bigg{[}\sum\limits_{i_{1}=1,\,i_{1}\neq
j}^{n_{1}}(\alpha_{1,i_{1}}-1)\,z_{1,i_{1}}(t)+\sum\limits_{i_{2}=1}^{n_{2}}(\alpha_{2,i_{2}}-1)\,x_{2,i_{2}}(t)\,+\\\
+\sum\limits_{j_{1}=1}^{m_{1}}\Big{(}{z}_{1,j_{1},0}(t)+(\varphi_{1,j_{1}}-1)\,z_{1,j_{1},1}(t)-\varphi_{1,j_{1}}\,z_{1,j_{1},2}(t)\Big{)}+\\\
+z_{1,m_{1}+1,0}(t)+(\varphi_{1,m_{1}+1,1}-1)z_{1,m_{1}+1,1}(t)+(\varphi_{1,m_{1}+1,2}-1)z_{1,m_{1}+1,2}(t)\,+\\\
+\sum\limits_{j_{2}=1}^{m_{2}}\Big{(}{x}_{2,j_{2},0}(t)+(\varphi_{2,j_{2}}-1)\,x_{2,j_{2},1}(t)-\varphi_{2,j_{2}}\,x_{2,j_{2},2}(t)\Big{)}\Bigg{]}+\eta_{2}(t)\,.$

The interior angles of the domain which is cut along the slit satisfy the
relation $\varphi_{1,m_{1}+1,1}+\varphi_{1,m_{1}+1,2}=\alpha_{1,j}$. The ODE
system for this family can be also rewritten in a suitable form.

##  5\. Examples

In this section we give some numerical examples showing the efficiency of our
method. For numeric calculations we used the Wolfram Mathematica package,
version 13.3.

######  Example 5.1.

Consider one parameter family of conformal mappings of annuli
$\mathcal{A}(t)=\\{\tau:\;q(t)<|\tau|<1\\}$ onto doubly connected polygonal
domains $\mathcal{D}(t)$, $0\leq t\leq 1$. The exterior boundary component of
every domain $\mathcal{D}(t)$ is the boundary of the fixed rectangle
$[-1,1]\times[-0.5,0.5]$, and the interior one is a rectilinear slit along the
segment $[a_{1}(t),a_{2}(t)]$ of the real axis (Fig. 5.1). We will assume that
the segment $[a_{1}(0),a_{2}(0)]$ coincides with $[-0.5,0.5]$ and the
endpoints of the slit move with constant velocities $v_{1}=\dot{a}_{1}(t)$ and
$v_{2}=\dot{a}_{2}(t)$. Therefore, we have $a_{1}(t)=v_{1}t-0.5$,
$a_{2}(t)=v_{2}t+0.5$, $0\leq t\leq 1$. The family of such conformal mappings
has the form

(35)
$\mathcal{F}(z,t)=C_{1}(t)\int\limits_{0}^{z}\exp\\{c\xi\\}\prod\limits_{i_{1}=1}^{4}\sigma^{-1/2}(\xi-
z_{1,i_{1}}(t))\,\prod\limits_{i_{2}=1}^{2}\sigma(\xi-
z_{2,i_{2}}(t))\,d\xi+C_{2}(t)\,,$

where $z=-i\ln\tau$ and $c(t)=\eta_{1}(t)/2+\eta_{2}(t)$. In (35) the
accessory parameters $z_{1,i_{1}}(t)$, $1\leq i_{1}\leq 4$, are symmetrical
with respect to the imaginary axis, i.e. $z_{1,1}(t)=-z_{1,4}(t)$ and
$z_{1,2}(t)=-z_{1,3}(t)$. The parameters $z_{2,1}(t)$ and $z_{2,2}(t)$ only
depend on the period $\omega_{2}(t)$, i.e.
$z_{2,1}(t)=\omega_{1}/2+\omega_{2}(t)/2$ and $z_{2,2}(t)=\omega_{2}(t)/2$.

Figure 5.1. The image of the polar grid under the conformal mapping of an
annulus onto the rectangle $(-1,1)\times(-0.5,0.5)$ with a slit along the
segment $[-0.1,0.5]$.

We note that earlier in Example 3.1, with the help of elliptic functions, we
found the values of parameters for conformal mappings of such domains. Here we
apply the results of Section 4 and determine the values of parameters at
$t=1$, considering that the initial values of the parameters (for $t=0$) are
known; we will take them from Table 3.1.

From Theorem 4.2 and Remark 4.2 it follows that the accessory parameters
satisfy the following differential equations

$\dot{\omega}_{2}(t)=i\sum\limits_{j_{2}=1}^{2}\mathcal{L}_{2,j_{2}}(t)\,,\quad\dot{z}_{1,k}(t)=-\sum\limits_{j_{2}=1}^{2}\mathcal{L}_{2,j_{2}}(t)\,\mathcal{K}_{2,j_{2}}(z_{1,k}(t),t)\,,\quad
1\leq k\leq 2\,,$
$\frac{\dot{C}_{1}(t)}{C_{1}(t)}=\sum\limits_{i_{1}=1}^{2}\mathcal{R}(z_{1,i_{1}}(t),t)-\sum\limits_{i_{2}=1}^{2}\mathcal{R}(z_{2,i_{2}}(t),t)-\sum\limits_{j_{2}=1}^{2}\mathcal{L}_{2,j_{2}}(t)\,\mathcal{P}({z}_{2,j_{2}}(t),t)\,.$

Solving the Cauchy problem for this ODE system, we obtain the values of the
accessory parameters and conformal modulus corresponding to the value $t=1$;
they are shown in Table 5.1 for different speeds of the endpoints of the slit.

Table 5.1. Some values of the accessory parameters and the conformal module for the conformal mapping of the annulus onto the rectangle $(-1,1)\times(-0.5,0.5)$ with a cut along the segment $[v_{1}-0.5,v_{2}+0.5]$. $v_{1}$ | $v_{2}$ | $\omega_{2}(1)$ | $z_{1,1}(1)$ | $z_{1,2}(1)$ | $m(1)$
---|---|---|---|---|---
$-0.25$ | $0.25$ | $1.5034301873176772\,i$ | $0.47104824707292203$ | $2.6705444064077497$ | $0.11963917295259123$
$0.25$ | $-0.25$ | $3.3301140782903405\,i$ | $0.1992807585816138$ | $2.9423118950085363$ | $0.2650020583099093$
$0.4$ | $0$ | $2.9794506900288944\,i$ | $0.39772855448531813$ | $3.0299504820063854$ | $0.2370971525083285$
$0.6$ | $0.4$ | $2.118321160564505\,i$ | $1.251124799561501$ | $3.099177418082569$ | $0.16857064188000076$

Comparison of the values from Tables 3.1 and 5.1 shows a good coincidence, up
to 9 digits after the decimal point. Numerical calculations show that the
accuracy of calculations falls when the endpoints of the slit are close enough
to the boundary of the rectangle, meanwhile the conformal modulus is very
close to zero.

######  Example 5.2.

Consider the problem of finding conformal mapping of an annulus
$\mathcal{A}=\\{\tau:\;q<|\tau|<1\\}$ onto the doubly connected domain
$\mathcal{D}_{1}$ that is obtained from the interior of the rectangle
$\Pi=(-1,1)\times(-0.5,0.5)$ by removing the triangle with vertices at the
points $w_{2,1}=0.25$, $w_{2,2}=-0.25$ and $w_{2,3}=0.25\,i$ (see Fig. 5.2).
By Theorem 3.1, the conformal mapping of $\mathcal{D}_{1}$ has the form

(36)
$\mathcal{F}(z)=C_{1}\int\limits_{0}^{z}\exp\\{c\xi\\}\prod\limits_{i_{1}=1}^{4}\sigma^{-1/2}(\xi-
z_{1,i_{1}})\,\sigma^{3/4}(\xi-z_{2,1})\,\sigma^{3/4}(\xi-
z_{2,2})\,\sigma^{1/2}(\xi-z_{2,3})\,d\xi+C_{2}\,,$

where

$c=\frac{\eta_{1}}{\omega_{1}}\Bigg{[}-\frac{1}{2}\sum\limits_{i_{1}=1}^{4}z_{1,i_{1}}+\frac{3}{4}\,x_{2,1}+\frac{3}{4}\,x_{2,2}+\frac{1}{2}\,x_{2,3}\Bigg{]}+\eta_{2}\,.$

Here $\mathcal{F}(z_{2,j})=w_{2,j}$, $1\leq j\leq 3$.

Figure 5.2. The image of the polar grid under the conformal mapping of the
annulus onto the domain obtained from the rectangle $(-1,1)\times(-0.5,0.5)$
by removing the triangle with vertices at the points $0.25$, $-0.25$ and
$0.25\,i$.

We will find approximate values of the parameters in (36). First we consider
the conformal mapping of an annulus onto the domain $\mathcal{D}_{0}$ that is
the rectangle $\Pi$ with a rectilinear slit along the segment $[-0.25,0.25]$;
all parameter for this mapping are known (Example 3.1). Then we will release
rectilinear slits from the points $w_{2,1}$ and $w_{2,2}$. Let the endpoints
of the slits move with constant velocities according to the laws
$v_{1}(t)=0.25(1+(i-1)t)$ and $v_{2}(t)=0.25(-1+(i+1)t)$. Then, at the moment
$t=1$ they meet at the point $w_{2,3}$. Denote by $\mathcal{D}(t)$ the doubly
connected domain that is obtained from $\Pi$ by making the slit along the
polygonal line with vertices at the points $w_{2,1}$, $w_{2,2}$, $v_{1}(t)$
and $v_{2}(t)$. We note that $\mathcal{D}(0)=\mathcal{D}_{0}$ and, as $t\to
1$, the family of domains $\mathcal{D}(t)$ converges to $\mathcal{D}_{1}$ as
to a kernel in the sense of Caratheodory (see [33, ch. 2, § 5; ch. 5, § 5]).

The family of conformal mappings of annuli
$\mathcal{A}(t)=\\{\tau:\;q(t)<|\tau|<1\\}$ onto $\mathcal{D}(t)$, $0<t<1$,
has the form

(37)
$\mathcal{F}(z,t)=C_{1}(t)\int\limits_{0}^{z}\exp\\{c(t)\,\xi\\}\prod\limits_{i_{1}=1}^{4}\sigma^{-1/2}(\xi-
z_{1,i_{1}}(t))\,\sigma(\xi-z_{2,1,0}(t))\,\sigma^{3/4}(\xi-z_{2,1,1}(t))\\\
\times\,\sigma^{-3/4}(\xi-z_{2,1,2}(t))\,\sigma(\xi-
z_{2,2,0}(t))\,\sigma^{-3/4}(\xi-z_{2,2,1}(t))\,\sigma^{3/4}(\xi-
z_{2,2,2}(t))\,d\xi+C_{2}(t)\,,$

where

$c(t)=\frac{\eta_{1}(t)}{\omega_{1}}\Bigg{[}-\frac{1}{2}\sum\limits_{i_{1}=1}^{4}z_{1,i_{1}}(t)+x_{2,1,0}(t)+\frac{3}{4}\,x_{2,1,1}(t)-\frac{3}{4}\,x_{2,1,2}(t)\,\\\
+\,x_{2,2,0}(t)-\frac{3}{4}\,x_{2,2,1}(t)+\frac{3}{4}\,x_{2,2,2}(t)\Bigg{]}+\eta_{2}(t)\,.$

Here the parameters $z_{1,i_{1}}(t)\;(\text{mod}\,\omega_{1})$, $1\leq
i_{1}\leq 4$, are real and taken in the ascending order on $[0,\omega_{1}]$,
and the parameters $z_{2,k,j}(t)=x_{2,k,j}(t)+\omega_{2}(t)/2$, $1\leq k\leq
2,0\leq j\leq 2$, are such that $x_{2,k,j}(t)\;(\text{mod}\,\omega_{1})$ are
real and taken in the ascending order on $[0,\omega_{1}]$. As we noted above,
for $t=0$ the values of the parameters coincide with the corresponding values
of parameters for the conformal mapping of an annulus onto $\mathcal{D}_{0}$;
we give them from Table 3.1, taking into account Remark 4.2.

Solving the Cauchy problem for the system of ODEs constructed for the family
(37) (see Theorem 4.2 and Remark 4.2), we obtain the needed values of the
accessory parameters and the conformal modulus of $\mathcal{D}_{1}$ at $t=1$;
they are given in Table 5.2. We also determine the values
$c(1)=-0.1455922673170743-1.2449331197330842\,i$ and
$C_{1}(1)=0.1299350302810949+0.11027609828928062\,i$.

Table 5.2. Values of accessory parameters and conformal module for mapping of an annulus onto a rectangle $(-1,1)\times(-0.5,0.5)$ with the triangle removed.  $m(1)$ | $\omega_{2}(1)$ | $z_{1,1}(1)$ | $z_{1,2}(1)$
---|---|---|---
$0.21029028897501653$ | $2.6425857078607464\,i$ | $0.185196737529923$ | $3.4688730938781616$
$z_{1,3}(1)$ | $z_{1,4}(1)$ | $x_{2,2,2}(1)$ | $x_{2,1,1}(1)$
$3.8320979743892862$ | $6.105157164198392$ | $6.109642627793988$ | $3.827612510793681$
$x_{2,1,0}(1)$ | $x_{2,1,2}(1)$ | $x_{2,2,1}(1)$ | $x_{2,2,0}(1)$
$1.8334660411846952$ | $1.8270349281572866$ | $1.8270349032507693$ | $1.8206037902233696$

As approximate values of the accessory parameters in (36) we can take
$z_{1,i_{1}}=z_{1,i_{1}}(1)$, $1\leq i_{1}\leq 4$, $z_{2,1}=z_{2,2,2}(1)$,
$z_{2,2}=z_{2,1,1}(1)$; for the value $z_{2,3}$ we can take either the
arithmetic mean of the numbers $z_{2,1,0}(1)$, $z_{2,1,2}(1)$, $z_{2,2,1}(1)$
and $z_{2,2,0}(1)$ or any of these values. We note that, as $t\to 1$, the
accessory parameters $z_{2,1,0}$, $z_{2,1,2}$, $z_{2,2,1}$ and $z_{2,2,0}$
approach each other indefinitely and merge into a unique point, but in
numerical calculations their values differ by values of the order of
$10^{-4}$; this is caused by the non-uniform convergence of the family of
conformal mappings near the specified point. Note that the accuracy of
calculations of polygon vertices was up to $10^{-6}$.

In Fig. 5.2 we show the polygonal domain $\mathcal{D}_{1}$ and the image of
the polar grid under the conformal mapping (36).

######  Example 5.3.

Consider the problem of constructing a conformal mapping of an annulus onto
the domain $\mathcal{D}=\Pi_{1}\backslash\Pi_{2}$, where
$\Pi_{1}=(-1,1)\times(-0.5,0.5)$ and $\Pi_{2}=[0.1,0.5]\times[0.1,0.2]$ (Fig.
5.3). The solution consists of four steps. On every step we solve the Cauchy
problem for a system of ODEs; the accessory parameters obtained on the
previous step, except for the final one, define the initial values of the
parameters on the next step. For convenience of presentation, we will assume
that on every step the parameter $t$ is on the segment $[0,1]$.

Step 1. As an initial, we take the conformal mapping of an annulus onto the
rectangle $\Pi_{1}$ with the slit along the segment $[-0.1,0.5]$; we take the
corresponding values of parameters from Table 3.1. We consider the family of
conformal mappings of annuli onto rectangle $\Pi_{1}$ with an elongating slit
along a two-link polyline composed from the segments $[-0.1,0.5]$ and
$[0.5,0.5+0.2t\,i]$.

Step 2. Now we consider the family of conformal mappings of annuli onto
$\Pi_{1}$ with the slit along the polyline consisting of three segments,

$[-0.1+0.6t,0.5],\ [0.5,0.5+0.2\,i]\ \text{and}\
[0.5+0.2\,i,0.5-0.4t+0.2\,i].$

We note that, with the growth of the parameter $t$, the first segment shortens
and shrinks to the point $0.5$ as $t\to 1$, and the third one grows; moreover,
it shrinks to the point $0.5+2\,i$ as $t\to 0$.

Step 3. On this step, the slit consists of three segments,

$[0.5+0.1t\,i,0.5+0.2\,i],\ [0.5+0.2\,i,0.1+0.2\,i]\ \text{and}\
[0.1+0.2\,i,0.1+0.2\,i-0.1t\,i].$

With the growth of $t$, the length of the first segment decreases from $0.2$
to $0.1$, and the length of the third one grows from $0$ to $0.1$.

Step 4. On the final step, the slit consists of three fixed segments
$[0.5+0.1\,i,0.5+0.2\,i]$, $[0.5+0.2\,i,0.1+0.2\,i]$,
$[0.1+0.2\,i,0.1+0.1\,i]$ and a growing one,
$[0.1+0.1\,i,0.1+0.1\,i+0.4t\,i]$. As $t\to 1$, the slit tends to the boundary
of $\Pi_{2}$ and, by the Caratheodory convergence theorem, the limiting
function maps an annulus onto the needed domain (Fig. 5.3). We also find the
values

$c(1)=-0.000055631445885073245-1.169959854131041\,i,$
$C_{1}(1)=-0.04708877454319929+0.12479568610390228\,i.$

The family of conformal mappings has the form

$\mathcal{F}(z,t)=C_{1}(t)\int\limits_{0}^{z}\exp\\{c(t)\,\xi\\}\prod\limits_{i_{1}=1}^{4}\sigma^{-1/2}(\xi-
z_{1,i_{1}}(t))\,\sigma(\xi-z_{2,1}(t))\,\sigma^{-1/2}(\xi-z_{2,2}(t))\\\
\times\,\sigma^{-1/2}(\xi-z_{2,3}(t))\,\sigma^{-1/2}(\xi-
z_{2,4}(t))\,\sigma(\xi-z_{2,5}(t))\,\sigma^{1/2}(\xi-z_{2,6}(t))\,\\\
\times\,\sigma^{1/2}(\xi-z_{2,7}(t))\,\sigma^{1/2}(\xi-
z_{2,8}(t))\,d\xi+C_{2}(t)\,,$

where

$c(t)=\frac{\eta_{1}(t)}{\omega_{1}}\Bigg{[}-\frac{1}{2}\sum\limits_{i_{1}=1}^{4}z_{1,i_{1}}(t)+x_{2,1}(t)-\frac{1}{2}\,x_{2,2}(t)-\frac{1}{2}\,x_{2,3}(t)\,-\frac{1}{2}\,x_{2,4}(t)\\\
+\,x_{2,5}(t)\,+\frac{1}{2}\,x_{2,6}(t)\,+\frac{1}{2}\,x_{2,7}(t)\,+\frac{1}{2}\,x_{2,8}(t)\Bigg{]}+\eta_{2}(t)\,.$

The values of the parameters for $\mathcal{F}(z,1)$ are given in Table 5.3.

As a result, we obtain that the conformal mapping of an annulus onto the
domain $\mathcal{D}$ has the form

$\mathcal{F}(z)=C_{1}\int\limits_{0}^{z}\exp\\{c\xi\\}\prod\limits_{i_{1}=1}^{4}\sigma^{-1/2}(\xi-
z_{1,i_{1}})\prod\limits_{i_{2}=1}^{4}\sigma^{1/2}(\xi-
z_{2,i_{2}})\,d\xi+C_{2}\,,$

where

$c=\frac{\eta_{1}}{\omega_{1}}\Bigg{[}-\frac{1}{2}\sum\limits_{i_{1}=1}^{4}z_{1,i_{1}}+\frac{1}{2}\sum\limits_{i_{2}=2}^{4}x_{2,i_{2}}\Bigg{]}+\eta_{2}\,.$

Here $z_{1,i_{1}}=z_{1,i_{1}}(1)$, $1\leq i_{1}\leq 4$,
$z_{2,1}=x_{2,1}(1)+\omega_{2}(1)/2$, $z_{2,2}=x_{2,6}(1)+\omega_{2}(1)/2$,
$z_{2,3}=x_{2,7}(1)+\omega_{2}(1)/2$ and $z_{2,4}=x_{2,8}(1)+\omega_{2}(1)/2$.

Figure 5.3. The image of the polar grid under the conformal mapping of an annulus onto the domain that is the difference of the rectangles $(-1,1)\times(-0.5,0.5)$ and $[0.1,0.5]\times[0.1,0.2]$. Table 5.3. Values of the accessory parameters and the conformal module for mapping of an annulus onto a domain that is the difference of the rectangles $(-1,1)\times(-0.5,0.5)$ and $[0.1,0.5]\times[0.1,0.2]$. $m(1)$ | $\omega_{2}(1)$
---|---
$0.22376354710663857$ | $2.8118956629256373\,i$
$z_{1,1}(1)$ | $z_{1,2}(1)$ | $z_{1,3}(1)$ | $z_{1,4}(1)$
$0.49730142210229983$ | $4.088563026275288$ | $4.217943920435873$ | $5.9070565839553115$
$x_{2,1}(1)$ | $x_{2,2}(1)$ | $x_{2,3}(1)$ | $x_{2,4}(1)$
$6.171115702928762$ | $6.16112894071675$ | $6.161128842028749$ | $6.1611288420232455$
$x_{2,5}(1)$ | $x_{2,6}(1)$ | $x_{2,7}(1)$ | $x_{2,8}(1)$
$6.157826022407041$ | $4.303856804569505$ | $3.408905071936863$ | $0.8254470359051237$

######  Example 5.4.

Using the method applied in Example 5.3 we also calculated accessory
parameters for the difference of two the rectangles,
$\Pi_{1}=(0,7)\times(0,4)$ and $\Pi_{2}=[a,c]\times[b,d]$ for different values
of $a$, $b$, $c$ and $d$. We note that such doubly connected domains
$\mathcal{D}$ were considered by H. Hakula, A. Rasila and M. Vuorinen [35]. We
also calculated conformal moduli $\text{Mod}(\mathcal{D})$ and capacities
$\text{Cap}(\mathcal{D})=(\text{Mod}(\mathcal{D}))^{-1}$ for such domains. In
Table 5.4 we give values of the moduli and capacities for some values of $a$,
$b$, $c$ and $d$. Comparison of our results with those from [35] shows a good
coincidence up to $10^{-9}$. We also note that for symmetric domains the
accuracy is higher, it is about $10^{-10}$.

Table 5.4. Some values of the conformal moduli and capacities of the domain $\mathcal{D}=\Pi_{1}\backslash\Pi_{2}$, where $\Pi_{1}=(0.7)\times(0.4)$ and $\Pi_{2}=[a,c]\times[b,d]$. | Our method |  Results from [35]
---|---|---
a$a$a |  a$b$a |  a$c$a |  a$d$a | $\text{Mod}(\mathcal{D})$ | $\text{Cap}(\mathcal{D})$ | $\text{Mod}(\mathcal{D})$ | $\text{Cap}(\mathcal{D})$
$1$ | $1$ | $2$ | $2$ | $0.1919267753916537$ | $5.210320435798281$ | $0.19192677723893617$ | $5.210320385649294$
$1$ | $1$ | $3$ | $2$ | $0.14823482589366233$ | $6.746053054478302$ | $0.1482348209832967$ | $6.746053277945276$
$1$ | $1$ | $4$ | $2$ | $0.1209178457885679$ | $8.270077865500184$ | $0.12091783807693261$ | $8.27007839293125$
$1$ | $1$ | $5$ | $2$ | $0.10139511296046277$ | $9.862408264093874$ | $0.10139510359023973$ | $9.86240917550835$
$2$ | $1$ | $3$ | $2$ | $0.21312544007028156$ | $4.6920724230304645$ | $0.21312544403732966$ | $4.692072335693745$
$2$ | $1$ | $4$ | $2$ | $0.16046010545976827$ | $6.232078666125066$ | $0.16046010434924893$ | $6.232078709256309$
$2$ | $1$ | $5$ | $2$ | $0.1277611567929334$ | $7.827105085004295$ | $0.12776115200936297$ | $7.827105378062926$
$3$ | $1$ | $4$ | $2$ | $0.21639757475743104$ | $4.621123878679977$ | $0.21639757713707608$ | $4.621123827863167$
$3$ | $1$ | $5$ | $2$ | $0.16046010585211312$ | $6.23207865088686$ | $0.16046010434924884$ | $6.232078709256313$
$4$ | $1$ | $5$ | $2$ | $0.21312544095079175$ | $4.692072403645554$ | $0.21312544403732897$ | $4.69207233569376$

##  6\. Discussion

There are many interesting questions concerning constructing conformal mapping
of polygonal domains. Our next problem is to consider one parameter families
of unbounded doubly connected polygonal domains and corresponding conformal
mappings. It is also interesting to increase the accuracy of calculation of
calculating accessory parameters and conformal moduli.

## FUNDING

This research is supported by the Russian Science Foundation, grant No
23-11-00066.

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