THEORETICAL CONSIDERATION OF CHARGE SIMULATION METHOD FOR NUMERICAL CONFORMAL MAPPINGS IN A RING DOMAIN

The charge simulation method has been applied to solve a lot of problems 
in electrical engineering. However, the principle of the method is not known enough even now. 
This paper is devoted to giving the theoretical and mathematical base for the charge simulation 
method of numerical conformal mappings in ring domains. Therefore for example, the uniform, convergence of approximations, the theoretical distribution of charge points, and the charges 
will be mathematically discussed. An example is shown to help understanding of theoretical 
considerations.


THEORETICAL CONSIDERATION OF CHARGE SIMULATION METHOD FOR NUMERICAL CONFORMAL MAPPINGS IN A RING DOMAIN 1. INTRODUCTION
The charge simulation method is very useful to solve partial differential equations in electrical engineering, and has been studied and developed by a lot of researchers.The method is easy to understand, and can be applied only by solving a system of simultaneous linear equations.Many examples show that the method makes possible to get rather precise solutions [or the boundary value problems with respect to domains bounded by smooth curves.However, many parts of the method depend on the experiment from the examles.For instance.
the best distribution of charge points is not known.
In this paper we will give the theoretical and mathematical base for the charge simulation method of numerical conformal mappings in ring domains.
Then, the potential theory [10,16], especially the asymptotic theorems [5-9,11-14] on extremal weighted hyperbolic polynomials play fundamental roles.The theorems depend on the notion introduced in 1992 by Mhaskar and Saff  [13].We note that numerical conformal mappings not depending on the charge simulation method have been shown in [4,15].
The results established in this paper are as follows: (la) A new scheme for approximations is proposed for the numerical conformal mapping of a ring domain.
(lb) The ditribution of charge points is characterized by the weighted extremal points.(lc) It is theoretically shown that the numbers of charge points in the complements of a ring domain must be equal to each another.
(ld) If the outer boundary is the unit circle with a center at the origin, the charge points exterior to it are represented by ( 1/,,, }=t, where {z,,, }= are the charge points interior to the inner boundary. (le) The approximations converge uniformly in the domain and on the boundary, if the charge points are theoretically distributed.
The outline of this paper is as follows.In sections 2 and 3, the definition of weighted hyperbolic capacity (shortly wh-capacity) and asymptotic lemmas on extremal wh-polynomials will be introduced, respectively.In section 4 the lemmas are applied to establishing algorithms of a new charge simulation method for numerical conformal mappings in ring domains whose each outer boundary is the unit circle with a center at the origin.In section 5 the theoretical distribution of charge poins is discussed.In section 6 a simple example is shown to help understanding of theoretical considerations for the new charge simulatin method.

DEFINITIONS
In approximation theory, the asymptotic behavior of extremal polynomials has been studied by a lot of researchers.In this section we describe the notions of weighted hyperbolic polynomials (shortly wh-polynomials) and weighted hyperbolic capacity (wh-capacity) introduced by the Let D be a ring domain whose outer and inner boundaries are the unit circle 9'0 with a center at the origin and a Jordan curve ", respectively.For the numerical conformal mapping of a ring domain we establish a new charge simulation method depending on asymptotic lemmas on extremal weighted hyperbolic polynomials.There the distance between z, and zj is defined by [lZ'-,z, z, I(z, )(=, ), where w(z) is a weight function.
All of the definitions for the usual weighted polynomials hold analogously for hyperbolic ones.
Let w w(z) be an arbitrary, continuous, positive function defined on the complex plane C. For each integer n _> 1, we let P,, denote the class of all polynomials of the form ,(z) (1-,z )()(,,)] (2.1) which we call wh-polynomials of degree n.
Let M{') denote the class of all positive unit Borel measures whose support is -.We define wh-capacity by where and caph(w, "),) exp(V,), ( / z-lw(z)w(t.)]da(z)da(t) sup I(a). (2.4) Throughout the remainder of this paper, we assume that every p,.,.,(z) has all of the zeros on ".
Though this assumption is rather strong, it is the key point to establish the new charge simulation method.
(2.5) The existence and the uniqueness of # were shown in [13,Theorem 3.1(b)].We assume that S 7, where S is the support of t.
For polynomials p,,(z) of degree n, the discrete unit measure defined on compact sets in the complex plane C with mass 1In at each zero of p,,,(z) will be denoted by/,, t(p,.).It will be called the normalized counting measure on the zeros of p,,,(z).If p,,,(z) has multiple zeros, the obvious modification will be considered.
The weak convergence of v to v as n x will be defined by lira /fdv,,--/fdv (2.7) for every continuous function in the complex plane C with compact support.

LEMMAS
We present the fundamental lemmas on extrmal wh-polynomials that are devoted to mathe- matical considerations of the new charge simulation method.
Under the assumption mentioned in section 2, we state the main lemma which has been verified in [6].
Then, there holds the equality lim ip,,(z)l '/' exp{g(z)} (3.1) uniformly on every compact subset of D, where (z) We note that if (a.1) holds uniformly on every compact subset of D, the property (ab) follows without inf and wih equality.It is easily shown by letting tend to 1 in (3.1).Now, we show the converse of Lamina a.1.Lemma 3.2.Let lit( v.,(z) . ,z)(z)(z-,')l be wh-polynomials of degree n satisfying the equa/ity im Iv.,(z)l '" x{(z)} (3.) uniformly on every compact subset o[D.
With the above notation we state a well-known theorem on the zeros distribution of extremal and bt, t(p,,,) be wh-polynomials of degree n and the normalized counting measure ofp, respectively, satisfying the equality lim i,(z)l / exp{g(z)} (3.4) uniformly on every compact subset of D. Then, there holds (A); g, converges weakly to In fact, Lemma 3.3 has been proved for the case when 7 lies on the real axis and for the w-polynomials =HI( _: . ,)()1. (3.5) Since the proof for the case of Lemma 3.3 is similar, we will omit it.
The following theorem has recently established by the author.
The ymptotic formula or the nth root of p,(z) can be obtained from one of ]p,(z)[.
Th follows from he fact that og(m,()), -og im,()l, + arg(p,(z)) '/ (3.13) is analytic in D and Cauchy-Riemann relation holds between the real and imaginal parts [14].Thus, using Lemma 3.6 with w(z) I/If(z)l the asymptotic formula for f(z) can be obtained from the nth root of the weighted polynomial s--I and caph(w, "y).
Lemma 3.7.Under the condit/on (A), the approximation f(z) of f(z) can be obtained by /() 'o{1-[ '' / (;) for a sull]ciently large number n.Then f,(z) converges to f(z) uniformly on every compact subset of D. 0 is determined by the condition f,(1) i.

ALGORITHMS
Using the terminologies of the charge simulation method, Lemma 3.7 is mentioned aa follows for the numerical conformal mapping of a ring domain.
Algorithm 4.1.The approximation f(z) of f(z) is obtained as follows: (4a) The charge points {z,,},__ satisfying the condition (A) are chosen on % (4b) The charge at every charge point is assumed to be 1/n.(4c) The approximation f(z) of f(z) can be obtained by Z-Zr,,t for a sufl]ciently large number n.Then f,(z) converges to f(z) uniformly on every compact subset of D.
is determined by the condition f (1) 1.
Algorithm 4.1 suggests us the following algorithm for the charge simulation method on the numerical conformal mapping of a ring domain.
Algorithm 4.2.The approximation f z) of f z) is obtained as follows: (4d) The charge points {z, }= are 'theoretically' chosen interior to ". (4e) The charge at every charge point is assumed to be 1/n.(4f) The approximation f(z) is represented by f,(z) e'"{H 1 ,,z }1/' (O;real) (4.2) t=l for a stdticiently large number n.Then f(z) converges to f(z) uniformly on every compact subset of D U ". O, is determined by the condition f(1) 1.
Algorithm 4.3 requires n-1 values I(,)1 within k 1,2,...,n to be given.However, all of the values are not required when the image domain D' is a disk with a center at the origin (o=ly the value II(C,)I-(k 1,2,...,,t) is =ot k=ow).From Algorithm 4.3 for this case, we have the following Algorithm 4.4.The approximation f(z) off(z) is obtained as follows: (4j) The charge points {z,,,}l'= and the collocation points {,,, },=, are appropriately chosen inteior to / and on % respectively.
The numerical experiments show that the approximations hold, when the charge points and the collocation points are theoretically distributed.The method introduced in this section has the following advantages compared to the standard one: (4m) The charge points 1 exterior to 70 is uniquely determined by ones {z.,, The number of charge points exterior to % is equal to one of them interior to -yp.
(40) When the charge points on the analytic curve whose image is a circle with center at the origin, typical examples show that numerical results of high accuracy can be obtain.

DISTRIBUTION OF CHARGE POINTS
In this section we show two kinds of weighted extremal points satisfying the condition (A).
For each integer n >_ 1, let Q. be a set of wh-polynomials where the zeros {z., },= lie on the boundary 7 of D. Let q,(z) be a polynomiM such that IIq2,( The existence of q,,,(z) is easily proved by the usual method.Then, q,,(z) is called a wh- Chebyshev polynomial with zeros on 7-It is known that it satisfies the condition (A) [5].
To show another polynomial satisfying he condition (A), we introduce the definition of wh- transfinite diameter.For each integer n > 9., let z,,,,-z,,,, lw(z,,.,)w(z,,,,)]}_/(,,(,,_,))In this section, a simple example is shown to help understanding of theoretical considerations or the new method introduced in the previous section.where D is a ring domain whose outer boundary is the unit circle.
Let f(z) map a ring domain D o bounded by % and a Jordan curve /p onto an annulus {w; 1 > Iwl > p} bounded by % and a Jordan curve 7'p, corresponding % to "p with p > 0.
The inverse function of f(z) is  w+a The charge points (z,, },1 on "r. and the collocation points {.,, },=1 on 7 are chosen as the images of the points pexp(2rj(i-1)), j v/-Z (6.4) being distributed on Iw] p under the mapping z h(w).We consider the case where p 0.1 and 0.6.It is known that these points are uniformly distributed (which means that these points satisfy the condition (A)) on the circles 7 and 7, respectively [3].
For n 10, the charge points with p 0.1 and the collocation points with p 0.6, we solve a system of simultaneous linear equations (4.6) and (4.7) and obtain the following results (to know the distribution precisely, all of the charges are denoted)-  The solution satisfies the relation (4.9) 1 a0--log0.6=-0.510825623..., a,--]-(i 1,2,..., 10) ( with high accuracy (it is remarkable that the number of charge points is only 10).
Using the charges {a,}'__ obtained above, the approximations (4.8) are represented.The accuracy of the errors are estimated by the maximum of If.(.,,+,/) (6.6) where ,,+/2 is the middle point between ,,, and ,,,+1.In fact, we must estimate the errors also at the collocation points.However, we can obtain anlogous results there.
Applying reflexion principle and maximum principle, the error on the unit circle will be estimated by one on the inner boundary %.Then, the accuracy may decrease by order of 10 -1 at most.We obtain the errors as follows (to know the distribution precisely, all of the errors are denoted):  Applying Algorithm 4.2 with n I0, using the above distribution of charge points and the theoretical values 1 (i 1:2, 10), (6.7) a0=log0.6,,=-6 the errors IJ' (z)-f(z)l of the approximation (4.2) are estimated.Then the errors are as follows: For the above example we have used the parameter p 0.1 for the charge points.If the charge points are not too near the boundary, analogous accuracy is obtained.

REMARKS
We have shown that the method introduced in this papar has the advantages copared to the standard one, when the charge points are theoretically distributed.
Amano's method [2] for the conformal mapping from a general ring domain to a standard one is applicable, even when the theoretical distribution is not known.It has also been shown that the approximations with high accuracy can be obtained.Algorithm 4.4 is applicable when the charge points and collocation points are distributed by the way of Amano.We wish that this paper would contribute to presenting the theoretical base also for Amano's method.