The determination of the poles of the mapping function and their use in numerical conformal mapping

Abstract

Let f be the function which maps conformally a simply-connected d o ma i n Ω o n t o t h e u n i t d i s c . T h i s p a p e r i s c o n c e r n e d w i t h t h e problem of determining the dominant poles of f in comp1(Ω∩∂Ω), and of using this information in order to obtain accurate numerical approximations to f by means of the Bergman kernel method.

The treatment of corner and pole-type singularities in numerical conformal mapping techniques

“…[26,Sect .4], the set {n j } must be chosen so that the resulting approximating series (3.1) converges rapidly. This can be achieved, as proposed in [22,26,29], by using an "augmented basis" formed by introducing into the "monomial set" z j -1 , j = 1,2,3,..., (3.2) functions that reflect the dominant singularities of f ' on ∂Ω and in I Ext (∂Ω)…”

“…Many numerical examples, illustrating the very considerable improvement in accuracy which is achieved by treating the singularities of the conformal maps in the manner described in earlier sections, can be found in references [3,14-16, 22,26-30]. (Of these [22,26,29] and [27] concern the use of the BKM and RM for the solution of problems P1 and P2 respectively, [3,28,30] the use of the ONM and VM for the solution of problem P3, and [14][15][16] Regarding the IEM, the method used in all examples is the collocation method of [15]. This method is based on approximating the density function v by cubic splines and "corner singular" functions, and it is described fully in [15,16].…”

“…This paper is a report of recent developments concerning the treatment of singularities in certain numerical methods for approximating the functions f I , f E and f D , which accomplish respectively the following three conformal maps. The paper is essentially a detailed survey of developments reported in [14,15,22,[26][27][28][29][30]. However, in Section 5 we also present certain new results that provide additional information about the singular behaviour of the interior and exterior mapping functions f I and f E .…”

“…In fact, this is the only geometry for which Levin et al [22] and Papamichael and Kokkinos [26] were able to determine the precise location and nature of the singularities of f I in Ext (∂Ω). Examples dealing with singularities corresponding to more general geometries can be found in [29] and also in and Ω corres-…”

Some Computational Aspects of a Method for Rational Approximation

“…We are interested in the shape of the If the locations of the singular points off(z) in fc are known, we try to distribute points ' so that a level curve L(t9) for some large t9 contains all the singular points off(z) in its interior. Sometimes the method of Papamichael et al [6] can be used to determine the location of the singular points closest to df. This information could also be used to introduce basis functions which are singular where f(z) is, see, e.g., Papamichael and Kokkinos [5].…”