Continued Fractions and Conformal Mappings for Domains with Angel Points

Here we construct the conformal mappings with the help of the continued fraction approximations. We first show that the method of [19] works for conformal mappings of the unit disk onto domains with acute external angles at the boundary. We give certain illustrative examples of these constructions. Next we outline the problem with domains which boudary possesses acute internal angles. Then we construct the method of rational root approximation in the right complex half-plane. First we construct the square root approximation and consider approximative properties of the mapping sequence in Theorem 1. Then we turn to the general case, namely, the continued fraction approximation of the rational root function in the complex right half-plane. These approximations converge to the algebraic root functions N z , N  N , C  z , 0 > Rez . This is proved in Theorem 2 of the aricle. Thus we prove convergence of this method and construct conformal approximate mappings of the unit disk onto domains with angles and thin domains. We estimate the convergence rate of the approximation sequences. Note that the closer the point is to zero or infinity and the lower is the ratio k/N the worse is the approximation. Also we give the examples that illustrate the conformal mapping construction.


Introduction
This article extends and develops paper [19]. There we presented the reparametrization method of conformal mapping of the unit disk onto the given simply connected domain with a smooth boundary. This method is based on reduction of Fredholm integral equation to a sufficiently large linear equation system and on the boundary curve reparametrization. The solution possesses polynomial form that can be easily analyzed. The method can be considered as one of the rapidly converging methods according to classification of [14]. The computation cost is actually similar to TheodorsenвЂ™s method or Fornberg method [10]. Let us compare the reparametrization method of [19] with the other conformal mapping methods. We do not consider the auxiliary mapping of the unit disk into subdomain of the given domain D as in the set of osculation methods [2]. The method of [19] does not require a sufficiently good initial approximation of the conformal mapping as the graphical methods such as that of [11]. The method does not apply any auxiliary constructions at the domain interior (domain triangulation [8], circle packing [15], domain decompositions, such as meshes of [24]). We do not need any iterative conformal mappings as in the zipper algorithm or the Schwartz-Christoffel mapping [7,13]. We construct our polynomial solution differently to the Fornberg polynomial method [9] that involves consequent approximations through suitable point choice at the domain boundary. Also we do not apply the solutions of auxiliary boundary value problems (the conjugate function method, Wegmann method [22,23]). Finally, the advantages of the method presented in [19] are the following: 1) it is devoid of auxiliary constructions, 2) it brings us to the mapping function in a polynomial form. The mapping function is a Taylor polynomial for the unit disk or a Laurent polynomial for the annulus in the case of multiconnected domains [1], [18]. Let us recall the basic construction steps of the reparametrization method [19].
The necessary condition for the function z z) ( ln  to be analytic in D is just as in [16] , and the unit disk is mapped to the domain bounded by the given smooth boundary ) (t z with the help of the Cauchy integral formula. So we construct an approximate polynomial conformal mapping. Similar method was also applied for construction of the annulus conformal mapping onto an arbitrary multiconnected domain with the smooth boundary in [1,17].
Note that we can reconstruct ) (t q intead of ) (t q in the case of smooth boundary [19]. So this method can also be considered as one of the methods using the derivatives [14]. The drawback of the reparametrization method is that it does not cover the conformal mappings of the unit disk onto domains with non-smooth boundaries. For instance, in the case of a domain with the angle  for at the point 0 t . In order to overcome this difficulty we apply the additional conformal mapping which "straightens" the bondary curve at the corresponding point. Then we apply the reparamentrization method to the new domain with the smooth boundary and again apply the conformal mapping that "bends" the boundary back to the initial one. Our aim is to represent this final mapping as the polynomial fraction. In the article we apply the modification of the conformal mapping construction of [19] both for domains with boundary angles and for slender regions. We present the mapping as a polynomial fraction. We first show that the method of [19] is applicable to domains with acute external angles. Then we present the polynomial fraction construction for the internal angle equal to /2  and conformally map the unit disk to the domain with such an angle. After that we construct the polynomial fraction for the angles Finally we show that this approach is valid for the conformal mapping of the unit disk to the slender region.

The case of an internal angle greater than 
The method of [19] allows us to solve the conformal mapping construction problem for any contour with the boundary curve forming internal angles greater than  . This can be illustrated , Chapter 2, formula (7.1)). In this case, can be made arbitrarily small for a sufficiently large n . That is, we have the convergence of the Fourier series at the angle point, regardless of the angle. This allows us to apply the conformal mapping construction method of [19]. Example 1. Consider the piecewise circular contour (two semicircles and one circle quarter) with the external angle /2  (Fig.1). First we approximate the boundary with a Fourier polynomial of degree 10 . Then we construct the approximating polynomial of degree 50 .  The similar example for the doubly connected domain with rectangular inner boundary can be found in [17].

The construction scheme for the case of an internal angle less than 
It is computationally difficult to apply the conformal mapping construction of [19] for a domain whose boundary forms an acute internal angle. Then the mapping polynomial converges slowly and the resulting conformal mapping angle point does not look like an angle at all (sort of a bubble). Consider a curve whose behavior at an angle point is similar to  (4), we consider the following inequalities: Thus, the method from [19] is difficult to apply, since even the Fourier series poorly approximate a curve with such an angle point. Let the domain boundary be angled and the angle be equal to  [5]. The most thorough and refined method here is the Padé rational function approximation of the algebraic function [3,4]. Note that these approximations are optimal in the set of fraction polynomials though their construction requires application of Euclidean algorythm and additional investigation of the holomorphness domain D .
The main result here is that the recursively constructed relations converge to the continued fraction approximating any rational The constructed sequence is clearly not Padé one. But the construction itself is fairly simple, does not possess nonunique solutions and provides convergence to the root function at the complex right half-plane. Similar results can be found in [5]. Also the author is sure that this result can be proved along the lines of [12]. Again the proof should apply induction and we need to consider the roots of the polynomials instead of the mapping itself. Note also that the fractional polynomial mappings can be applied, for instance, to exact solution of the elasticity theory problems [20].

The square root approximation
First consider the basic problem of the square root fraction polynomial representation. It is well-known that . This gives rise to the following recursive procedure: The proof is by induction.
. The induction step is as follows: Consider    This completes the proof. Assume now that we have a convex domain with acute internal angles and we need to construct the conformal mapping of the unit disk onto this domain. The main construction steps are as follows: we make the domain as round as possible with square mappings. If the resulting domain does not overlap itself then we construct the approximating polynomial according to the method of [19]. Finally we construct the square root approximations of the resulting image inverse to the squares of the first step. Example 3. Let us construct an approximate conformal map of the unit disk onto the contour with the internal angle /2  . Here we have the 11th iteration of the square root approximation and degree 50 polynomial for the initial domain (Fig.3).
The third approximation then equals The relation we need then equals So the more acute the angle and the closer z is to 0 the worse is the approximation convergence. This completes the proof of the theorem.
We now construct the following mappings exactly as in Example 3.
The unfolded domain was approximated by the polynomial of degree 50 . We next apply the 6th fraction iteration to fold the domain back to the angled one (Fig.4).  These examples show us that the more acute is the internal angle the harder it is to approximate it.

The case of thin domains
Consider the case of slender regions. The second problem for us is the case of relatively thin domains (e.g. ellipse with two significantly different axes). Consider the integral equation of [19] kernel behavior for  close to the point t of the largest possible curvature Then the diagonal elements of the relative linear equation system matrix are close to ) (t  and are also large. Thus, the greater the curvature ) (t  of the curve in t , the worse the convergence of the polynomial solution. The authors of [6] numerically solve the singular integral equation in order to find the conformal mappings from elliptic to slender regions. The method of recursive fractions is also applicable to the conformal mapping construction of a disk onto a thin domain. The main problem here is the so-called point crowding phenomenon. Here we achieve the similar results (domain sides ratio 1/4 ) with our method as a natural application. We first make the domain less slender with the help of the square mapping 2 ) ( a z  , here the point a lies outside the domain and close to its boundary point of maximal curvature. We cannot take this point at the boundary itself since then we achieve the domain that cannot be immediately inserted into the right half-plane at the neighbourhood of a . Secondly we apply the approximate conformal mapping construction algorithm. Finally we apply the square root approximation in order to return to the domain with the given boundary. Now, if a domain lies between two sides of the right angle closely to the vertex then we consider the mapping of the disk onto the squared domain and the square root approximation of the angle. Example 6. Consider the ellipse of semiaxes 1 and 1/4 : 1 = 16 2 2 y x  . Let us construct an approximate conformal mapping of the unit disk onto this ellipse. The initial method of [19] provides us with the following result for the polynomial of degree 1200 (Fig.6). Here we consider the 20th square root iterations and 1000 degree polynomial (Fig.7). Similar picture under only polynomial approximation due to the point crowding phenomenon happens for polynomial of degree 4 10 .

Conclusion
We first showed that the method of [19] works for conformal mappings of the unit disk onto domains with acute external angles at the boundary. Next we outlined the problem with domains which boudary possesses acute internal angles. Then we constructed a method of rational root approximation in the right complex half-plane. Also we proved convergence of this method and constructed conformal approximate mappings of the unit disk onto domains with angles and thin domains. All the constructions of the article are supported by examples.
Our approach of continuous fractions application to conformal mapping constructions shows good convergence and may be appied, for example to certain problems of mathematical physics, particulary, to elasticity theory problems.