Radial and circular slit maps of unbounded multiply connected circle domains

T.K DeLillo () T.A Driscoll A.R Elcrat J.A Pfaltzgraff 0 Department of Mathematics, University of North Carolina , Chapel Hill, NC 27599 , USA 1 Department of Mathematical Sciences, University of Delaware , Newark, DE 19716 , USA 2 Department of Mathematics and Statistics, Wichita State University , Wichita, KS 67620-0033 , USA Articles on similar topics can be found in the following collections Receive free email alerts when new articles cite this article - sign up in the box at the top right-hand corner of the article or click here - Subject collections Email alerting service To subscribe to Proc. R. Soc. A go to: http://rspa.royalsocietypublishing.org/subscriptions Radial and circular slit maps of unbounded multiply connected circle domains Infinite product formulae for conformally mapping an unbounded multiply connected circle domain to an unbounded canonical radial or circular slit domain, or to domains with both radial and circular slit boundary components are derived and implemented numerically and graphically. The formulae are generated by analytic continuation with the reflection principle. Convergence of the infinite products is proved for domains with sufficiently well-separated boundary components. Some recent progress in the numerical implementation of infinite product mapping formulae is presented. 1. Introduction We develop formulae for conformal maps f :U/P from unbounded multiply connected circle domains to canonical unbounded slit domains. A circle domain U is a domain of connectivity m in the extended complex plane C that contains the point at infinity, and whose m boundary components are circles, Cj, jZ1, ., m. A radial or circular slit domain P is a domain in C , N2P with boundary consisting of m closed segments lying on rays from the origin or m closed circular arcs lying on circles centred at the origin, respectively. Circle, radial slit and circular slit domains are three of the classes of canonical domains in Koebes classification of multiply connected domains. There are various functional relationships between pairs of slit mappings from different canonical classes ( Nehari 1952, Chap. 7), but the circle domains are not related to other canonical classes in such an elementary fashion. Thus, it is of great interest to be able to find explicit formulae for mapping the circle domains onto the radial and the circular slit domains. We derive our mapping formulae by using the reflection principle to extend the mapping f beyond U to a globally defined function. Then, complete knowledge of the zeros and poles of the globally defined function enables one to express f as an T. K. DeLillo et al. infinite product. This is a more direct determination of f than the analogous process in finding the SchwarzChristoffel mapping formula for general polygonal domains where the reflection process leads to a determination of the derivative of the mapping function (DeLillo et al. 2004). The remaining problem of trying to determine f from an integral of many non-elementary infinite products with unknown accessory parameters is still not solved in a satisfactory general manner (DeLillo et al. 2006). Thus, it is quite interesting to see in the present work that for radial and circular slit mappings, the problem of integrating the derivative of the desired mapping function is eliminated. SchwarzChristoffel formulae for bounded polygonal domains were derived by Crowdy (2005) and for unbounded polygonal domains by Crowdy (2007) using SchottkyKlein prime functions (see also DeLillo 2006). The techniques in this paper differ from those employed by Crowdy & Marshall (2006). They follow the approach of Schiffer (1950) giving the radial and circular slit maps in terms of Greens functions by using SchottkyKlein prime functions of the circular domains. By contrast, we use directly the properties that such mappings must have and basic reflection arguments to derive our formulae without recourse to Greens functions. There is a close connection between reflections in circles and the Schottky group, which DeLillo (2006) uses to derive relations between the SchottkyKlein prime functions and (bounded and circular) slit maps in the context of SchwarzChristoffel mapping. Greens functions for multiply connected domains are useful in many applications. Crowdy & Marshall (2007) have given Greens functions for circle domains in terms of SchottkyKlein prime functions. Our methods can also be used to give explicit formulae for Greens functions for circle domains. As given by Nehari (1952), the radial and circular slit maps are central components in the construction of Greens functions of a given domain. In a similar fashion, the combined circular/radial slit map given below can be used for the construction of the Robin function, the Greens function for the mixed boundary-value problem. With our approach, the maps to (bounded) circular and radial slit discs and annuli are also needed. However, these maps are closely relat (...truncated)