CC BY
71DOI: 10.15393/j3.art.2018.5411
The paper is presented at the conference "Complex analysis and its applications" (COMAN 2018), Gelendzhik - Krasnodar, Russia, June 2-9, 2018.
UDC 517.542
I. A. KOLESNIKOV
ON THE PROBLEM OF DETERMINING PARAMETERS IN THE SCHWARZ EQUATION
Abstract. P. P. Kufarev’s method makes it possible to reduce the problem of determining the parameters in the Schwarz-Chris-toffel integral to the problem of successive solutions of systems of ordinary differential equations. B. G. Baibarin obtained a generalization of this method for the problem of determining parameters (preimages of vertices and accessory parameters) in the Schwarz differential equation, whose solution is a holomorphic univalent mapping from the upper half-plane onto a circular-arc polygon. This paper specifies the initial condition for the system of differential equations for the parameters of the Schwarz equation obtained by B. G. Baibarin. This method is used to solve the problem of determining the accessory parameters for some particular mappings.
Key words: conformal mapping, Schwarz equation, accessory parameters, parametric method, circular-arc polygon
2010 Mathematical Subject Classification: 30C20
1. Introduction. There is a classical approach for constructing a conformal mapping from the canonical domain (unit disk or complex half-plane) onto a circular-arc polygon based on the Schwarz differential equation. The well-known problem of determining the parameters of this equation is solved for certain particular mappings onto circular-arc polygons. The simplest and most studied cases are when the circular-arc polygon has no more than three vertices [7], [14]. The parameter problem is solved for some more complicated particular cases on the basis of the P. Ya. Polubarinova-Kochina method in works by P. Ya. Polubarinova-Kochina, E. N. Bereslavsky and others, see for example [4] and the overview work [3]. We also mark the approach of A. R. Tsitskishvili, based on the
©Petrozavodsk State University, 2018
theory of conjugation for several unknown functions and on the theory of I. A. Lappo-Danilevskii [17], [18], which allows us obtaining some particular cases. However, it is oriented to general studies, as well as the work of L. I. Chibrikova (for example [5]).
These papers are of interest both for constructing conformal mappings and for studying differential equations of the Fuchs class. The conformal mapping of a canonical domain onto a polygon with a boundary consisting from segments can be represented by the Schwarz-Christoffel integral. The problem of determining the parameters of these mapping is simpler, since for an n-polygon with a straight-line boundary it is sufficient to define n preimages of vertices, while for a circular-arc n-polygon it is necessary to define 2n parameters: n preimages of vertices and n additional parameters, called accessory ones. In 1947, P. P. Kufarev [15] (see also [1], [16]) proposed a method for defining preimages of vertices in the Schwarz-Christoffel integral for mapping from the unit disk onto a polygon with internal normalization. For special cases, the method was first tested in the work of Yu. V. Chistyakov [6], then in was applied in [9]. The method is convenient for practice, it received various generalizations. Thus, the method is extended to mappings with boundary normalization in the work of V. Ya. Gutlyansky and A. O. Zaidan [8]. L. Yu. Nizamayeva [12], [11] proposed a new approach of finding the parameters in the Schwarz-Christoffel integral, using the idea of P. P. Ku-farev and the technique of Hilbert boundary value problems with piecewise smooth coefficients and variations of such problem solutions. In the work of N. N. Nakipov and S. R. Nasyrov [10], the method is generalized to mappings onto multisheet polygons containing branch points. In [13] Ku-farev's method is extended for mappings from a half-plane onto numerable polygons with transfer symmetry. B. G. Baibarin generalizes P. P. Ku-farev's method for the problem of determining parameters in the Schwarz differential equation, representing a holomorphic and univalent mapping from the upper half-plane onto a circular-arc polygon. This paper specifies the result of B. G. Baibarin [2]. With help of the generalization obtained by B. G. Baibarin we define accessory parameters for some particular mappings.
2. On Kufarev’s method. We briefly describe P. P. Kufarev’s method. Suppose we need to obtain a conformal map from the upper half-plane onto some circular polygon Д. Without loss of generality, we can assume it contains the origin and that the polygon Д is a kernel with respect to the origin of some family of simply connected domains
A(t), 0 < t < T. Here the family A(t) is obtained by carrying out a cut along N arcs of circles in some initial domain A0, A(0) = Ao, and A(T) = A. There is a family of functions f = f (z,t) that maps the upper half-plane П+ = {z G C : Im z > 0} onto A(t). In the first step, carrying a cut along the first arc, we have a family of mappings f = f (z,t), represented by Schwarz’s differential equation on the one hand, and the Loewner equation on the other. Note that the parameters of the map f = f (z,t) (the preimages of the vertices of the polygon A(t) and the accessory parameters) change continuously with the length of the cut. Using the differential equations of Loewner and Schwarz, one can obtain a system of ordinary differential equations for the parameters of the map f. In the paper of P. P. Kufarev [15], devoted to the determination of parameters in the Schwarz-Christoffel integral, the initial conditions of the ODE system at the first step are the parameters of the map f = f (z, 0) (the initial domain A0 can be chosen sufficiently simple to write explicitly the map f = f(z,0)). In the generalization of P. P. Ku-farev’s method to the case of mappings onto circular-arc polygons, additional difficulties arise in determining the initial conditions of the ODE system (for more about this, see the following paragraphs). Let us integrate the ODE system for the corresponding value of the parameter t = ti, that is, we define the parameters of the function f = f (z,U) that maps the upper half-plane onto the domain A(t1) (the domain A0 with a cut along the first arc). Then we can proceed to the second step and carry out the cut along the second arc and define the parameters of the corresponding family of functions. Thus, in N steps, we can define the parameters of the map f = f(z, T).
3. The main results of B. G. Baibarin generalization. In this section we present the main results of the work [2].
Let L(t) = {Z : Z = Z(т),ti < т < t}, 0 < ti, t < tn+i, Z(tp) = Zp, p = 1,..., П, Z1 > 0, be a piecewise smooth curve consisting of circular arcs that does not pass through the origin (n is an even number). Denote by A(t) a domain, obtained from the plane by carrying out a cut along the positive part of the real axis from the point Zi to infinity and excluding the curve L (the curve L and the cut intersect only at the point Z1), A(t) = C\(L(t)U{Z G C : Re Z > Z1, Im Z = 0}). Let the family of functions w = w(z,t), t1 ^ t ^ tn+1, map the half-plane onto the family A(t) (Fig. 1), such that w(e(t),t) = 0, where в satisfies the differential
equation
de(t)
dt
1
в(t) - A(t)’
e(t 1) = в\.
(1)
Here Л(t) is the preimage of the movable end of the cut Z(t). Let the infinity be fixed under the mapping w(z,t) and w(^,t) maps points Лn+ 1-p(t), Лn +p(t) in point Zp, p = 1,..., П. At the vertex, whose preimage is the point ap, we denote the inner angle of D(t) by apn. Therefore, we have ap = 2 — a„_p+1. Note that we can choose any convenient domain [1] as
the initial region Д(0).
Denote the Schwarz derivative of the mapping w by S, S(w,t):= w"'(z'f) 3 fWiPA
w'(z,t) 2 w'(z,t)
A family of mappings w = w(z,t) satisfies the Schwarz differential equation
S(z,t) =
p=0 1
Lv
Mp(t)
(z — Лp(t))2 z — Лp (t)
+
(2)
where i\(t) = ao(t), Lp = -(1 — ap).
Since w satisfies the Loewner differential equation in the half-plane
dw(z,t) 1 dw(z,t)
dt z — Л^) dz
the Schwarz derivative S satisfies the equation dS(z,t) 1 dS(z,t) 2S(z,t)
= 0, w(z,ti) = Zi + z2,
dt
z — X(t) dz (z — Л(^)2 (z — X(t))4
= 0 (3)
6
with the initial condition S(z,to) = — fут.
The function on the left-hand side of (3) has poles of the third order at the points z = AP, p = 1,... ,n. At the point z = Ao = A it has a pole of the forth order. On the other hand, in the right-hand side of the equality (3) the function is identically equal to zero. It follows that the parameters AP, MP, p = 0,1,..., n, of the Schwarz derivative S of the function w satisfy the system of ordinary differential equations
1
aP(t) = у ,
ap(t) — A (t) A'(t) = —Mo(t),
MP(t) + Lp (aP(t)) -
Em; (t) = 0.
„ P=0
p = b ..^n
(Ap(t))2 Mp(t)
0, p = 1,...,n,
(4)
Next, we make the change of variable x2 = t—tn. Then the parameters
Ap(t) = Ap(x2 —tn) = ap(x), Mp(t) = Mp(x2 —tn)
MP(x) p = 1,. ..,n,
A(t) = A(x2 — tn) = A(x), Mo(t) = Mo(x2 — tn) = Mo (x), as functions of the variable x are expanded in series ao(x) = A(x) = a + Aix + A2x2 + A3x3 + ..., ap(x) apo + apix + ap2x + ap3x + . . . , p 1, . . . , n,
Mp(x) = mpo + mpix + mp2x2 + mp3x3 + .. ., p = 2,. .., n — 1, (5)
Mp(x) = mpx 1 + mpo + mpix + mp2x2 + . .., p = 0, 1, n,
and satisfy the system of differential equations obtained from the system (4) by changing of variable t = x2 — tn:
aP(x)
A'(x)
2x
ap(x) — A(x), p —2xMo(x),
^ ... ^
< 2x2Mp(x) + Lp (ap(x))3 — x (ap(x))2 Mp(x)
n
E M^ (x) = 0.
„ P=o
0, p = 1,...,n,
(6)
The values «ю = ano = a, apo, mpo, p = 2,... ,n — 1, are known if the mapping w = w(z,t n) is specified.
There are the following particular integrals of the system (4)
Y^Mp = 0,
p=0
(7)
^ ^ apMp
p=0
n
—2 — £ Lp
p=0
M0 +2 £
p=i
Lp
Mp
(«p — A)2 («p — A)3
0.
Substituting the expansions (5) into the system (6), for the coefficients Ai, «pi, p = 1,..., n, mo,-i, mi,_i, mn,-i, mp,i, p = 2,..., n — 1, we obtain a system of algebraic equations that has a real solution (ai,an = 0, 2)
__ /о an _ /0 ai \ _ I
aii — \ 2 , ani — \ 2 , Ai — aii + anl,
ai V an
ni
>, mn,_ i Ln
= Ai = т
m0,-i = 2 , mi,-i = Li "a2i +2, mn,-i = Ln ani + 2,
api =0, p = 2,. .., n — 1, mpi =0, p = 2,. .., n — 1.
(8)
To determine 2n + 2 second coefficients of the series (5) we get 2n +1 linearly independent equations
aii
1
\ ii \ ni
ai2 = A^——ту-, an2 = A^——ту-, ap2 -
4 + a-|_i 4 + ani a,po
-, p = 2,...,n —1,
moo = — A2, mio = 2Liai2 ii '™ - OT - - ni
■11 + 2
mno — 2Lnan2“
n i + 6
n i +2,
„ mp2 ap2 ( mpo 2Lpap^, p 2, ...,n 1.
(9)
To determine 2n + 2 of the third coefficients, we get a system of 2n + 1
2
— a
linearly independent equations
8A2a11
«13 —
+
ll
(all + 4)2(a1i + 6) a2i + 6
8A2a3
2an1
"3 (ah + 4)2 (аП,1 + 6) ah +6
3 '
2
2 A1
a2
+ У^ Аз,
«p3
m01 0 A3, m11 3L1 4 A3, m„i 3Ln 4 . A3
2 a4! - 4 <1 - 4
о P — 2,...,n - l,
3 (ap0 a)2
. mp3 2ap2 ap3 ( mp0 3Lpap2^j, p 2,...,n 1.
(10)
2
To determine the fourth and subsequent coefficients apk, p — 1,..., n, m0,k—2, m1k—2, mn,k—2, mpk, p — 2, ...,n — 1, substitute the expansions (5) into the system (6) to get 2n + 2 linearly independent equations, k — 4,5,.... Thus, the fourth and subsequent coefficients in the expansion (5) are determined by the series method. In [2], the convergence of the series (5) whose coefficients are found by this method is proved. Specifying the initial conditions of a system of ordinary differential equations (6) (finding the coefficients A2 and A3) requires additional effort.
4. Addition to B.G. Baibarin’s results. It is not possible to find the coefficients A2 and A3 with the help of the work [2].
Substituting the series (5) into the equality (7) and equating the free terms we obtain
n
УЬ mpo — 0,
p=0
that is
A2
(2 + a1)(2 + an)
9a1an
n—1
y^m-pp.
p=2
(11)
The coefficient A3 depends on the curvature of the arc, along which the cut at the current step is carried out, but the relationship between A3 and the curvature of the arc is not established.
Note that the expansion of the Schwarz derivative S of the function w at infinity has the form 1
S(w,z)
1 — + bi + b2 +
2z2 + z3 + z4 + ...:
where aTOn = —2n is angle at infinity, 61,62 are real constants. So one can write [14] the equation (2) as
S{w,z) = YJj—
Lp
p=0
(z — ap )2
+
n \ n-2
2 — n — a200 + E a2 zn-1 + E Yvzv
j=0 J v=0
n ,
2 П(z — aj)
j=0
here yv are some real parameters. Now we can turn from n +1 unknowns parameters M0, M1,..., Mn to n — 1 unknown parameters y0,..., yn-2 by the formula
Mp
2 — n — a^ + E a
n
ak
n-2
+ E Yv ak v=0
1
n ’
2 П (ak — aj)
j=0,j=k
P
0,...,n.
(12)
5. A particular cases. Let us consider the particular case when n = 2. The function w = w(z,t) maps the upper half-plane onto a circular-arc polygon Д(t), which is a plane with a cut along the ray from the point z1 > 0 to infinity and with a cut along the arc of the circle starting from the point Z1. There are four preimages of vertices under the mapping w, they are A, a1,a2 and infinity, a2 ^ A ^ a1, w(<x,t) = to; the angles at the corresponding vertices are equal to 2n, an, (2 — a)n, —2n. We note that, according to the chosen normalization (1) f (z,0) = z2 + Z1. The mapping w = w(z,t) satisfies the Schwarz differential equation. With the help of the formula (12), the accessory parameters of this equation can be written in the form
= 2A(1 — a)2 + Y0 2(A — a1 )(A — a2) ’
M1
2a1(1 — a)2 + Y0 2(a1 — A)(a1 — a2) ’
M = 2a2(1 — a)2 + Y0 2 2(a2 — A)(a2 — a1).
The ODE system (6) takes the form
aP(x) = —,, p =1, 2,
A'(x) = —2xMo(x),
2x2Mp(x) + Lp (a'p(x))3 — x (a'p(x))2 Mp(x) = 0, p = 1, 2,
n
E Mp(x) = o,
p=0
hence, taking into account that m0i = — 1.5A3, we obtain
Yo(x) = —2(ai(3 — a) + a2(1 + a) — A(3 — a)(1 + a)). The system of ODE can now be written in the form
1
a/p(T)
ap(T) — A(t )’
P = 1, 2,
A'(r) = —
where т = x2, or as follows dai(A)
2A(t )(1 — o;)2 + Yo (т)
2(A — ai(T))(A — a2(T)) ’
A — a2(A)
dA 4A — ai(A)(3 — a) — a2(A)(1 + a) ’
da2(A) A — ai(A)
dA 4A — ai(A)(3 — a) — a2(A)(1 + a)
After the substitution
p(A) = (1 + a)(a2(A) — A) + (3 — a)(ai (A) — A),
q(A) = (3 — a^a2 (A) — A + (1 + a) (ai (A) — A, the system becomes simpler:
' dP(A) _ q(A)
dA p(A)
dq(A)
4,
dA
= -3.
Hence, taking into account the normalization ai(0) = a2(0) = A(0) = 0, conditions (8) and conditions (9), (10), (11), which in this case take the
2 — a a .
form A2 = ai2 = a22 = 0, ai3 = , —т A3, a23 = ^-77.------г A3, we obtain
2(1 + a) 2(3 — a)
A = ai(1 + a) + a2(3 — a),
(a2 — ai)3 = c(a2(2 — —) + a1—),
where
c
8y/2a(2
—-(— — 3)(1 + a) a) (2 - a)3a3A3 '
Consider the behavior of the parameters a1 = a1(x), a2 A = A(x), using their expansions (5), for fixed A3 and —.
a2 (x),
Figure 2: The graphs of a1, a2 and A
Let — = 0.5, A3 = —1. Then x G [0; X], X « 2.0205, and lim A(x) = = lim a2(x) « —3.28, lim a2(x) « 3.28. Graphs of functions a1, a2 and
A are shown in Fig. 2a) (the graph of a1 is on the top, the graph of A is the middle one, and the graph of a2 is on the bottom). Hence, we conclude that this case corresponds to the mapping w from the half-plane onto the circular-arc polygon shown in Fig. 3a).
Let — = 0.5, A3 = 1. ThenxG [0; X], X«0.869, and lim^a2(x)«—0.634,
lim A(x) = lim a1(x) « 3,14. Graphs of functions a1, a2, A are shown in
Fig. 2b) (the graph of a1 is on the top, the graph of A is the middle one, and the graph of a2 is on the bottom). Hence we can conclude that this case corresponds to the mapping w from the half-plane onto the circular-arc polygon shown in Fig. 3b).
Let a = 0.5, A3 = 0. Then a1 , a2 and A are linear functions of the variable x, x G [0;+to), their graph are shown in Fig. 2c). This case corresponds to the mapping w from the half-plane onto the polygon shown in Fig. 3c).
Next, we carry out another cut (n = 4) and let
A3 =
____________^ — V Ы-2 — 460 — 94—
81u2(2 — 3k — —)2(3k + —)V(2 — в)3в3 ^
——2(13 — 8в + 4в2) + 2—(13 — 8в + 4в2) —
—9k2(133 + 22в — 11в2) + 3k(1 — —)(133 + 22в — 11в2^),
a) A3 = -1
b) A3 = 1
Figure 3: The range of w
where u = a3(0) — a2(0), k = —, в = «1, a = a2. We have five preimages
c
of the vertices a3(x) ^ a4(x) ^ A(x) ^ a1(x) ^ a2(x) under the mapping w. Note that A(0) = a1(0) = a4(0), a3(0), a2(0), c have the values defined in the previous step-
The parameters Mp, p = 0,1, 2,3,4, can be written according to the formula (12):
Y0 = (a — 3)(1 — в )Aai a,2 — (1 + a)(1 — e)Aaia3 + (3 — a)(1 + a)Aaia4—
— (1 — e)2Aa2a3 + (3 — a)(1 — e)Aa2a4 + (1 + a)(1 — e)Aa3a4+
+(1 — e)aia2a3 — (3 — a)aia2a4 — (1 + a)aia3a4 — (1 — e)a2a3a4,
-2 — Aai ((1 — a)2 — 4в) + Aa2 (1 — в)2 + Aa3(1 — в)2+
+Aa^ (1 — a)2 — 4(2 — в)) + aia^ 5 — 2в — a(2 — в)) + +aia^ 1 + a(2 — в)) + aia4(1 — a)2 + a2a3(1 — в )2+ +a2a4 (1 + в(2 — a)) + a3a4(1 + aв),
Y2 = A 1(3 — a)(1 + a) — (1 — в )2) —
ai
((1 — a)2 + 3(1 — в ^ +
+a2 ( a — 4 + в(2 — в)) + a3 ( в (2 — в) — 2 — a) + a4 ( 3(1 — в) — (1 — a)2).
Indeed, it can be verified directly that Mp satisfies the system (6) and the conditions (8), (9), (10). Consider behavior of the parameters ai, a2, a3, a4 and A for the chosen A3 and a = 0.5, в = 1/3. Suppose that, for the first arc, A3 = —1 and x = 1. Then A(0) = ai(0) = = a4(0) « 0.87689, a3(0) « —0.91502, a2(0) « 2.10963. For the second arc, we have x G [0;X],X « 0.3647, and lim A(x) = lim ai(x) =
= lim a2(x) « 2.4502, lim a3(x) « —0.9677, lim a4(x) « 0.6618. Func-
x^X x^X x^X
tion graphs of a2, ai, A, a4 and a3 are shown, from top to bottom, in
2
a) Behavior of the parameters b) The range of w
Figure 4: Second step
Fig 4a). In this case, we see that, as x tends to X, the second arc approaches to a point of the ray, as shown in Fig. 4b).
Acknowledgment. This work was supported by RFBR according to the research project 18-31-00190\18.
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Received May 31, 2018.
In revised form, August 30, 2018.
Accepted September 1, 2018.
Published online September 17, 2018.
Tomsk State University
36 Lenina pr., Tomsk 634050, Russia
E-mail: [email protected]
