2026/07/01 by Klaus Schiefermayr, Olivier Sète
Mathematics · #Limits and Structures in Graph Theory #Mathematical Dynamics and Fractals #Mathematics and Applications #math.CV #msc:30C20 #msc:30C35 #msc:31A05 #msc:33E05
paper · pdf · doi:10.1016/j.jmaa.2026.130976
arxiv created 2026/07/31 · arxiv updated 2026/08/03
The aim of this paper is threefold: (i) We study the complex Green's function (the analytic extension of the real Green's function) for multiply connected domains with some symmetry and transform it to a simple form with the help of Walsh's conformal map onto lemniscatic domains. (ii) For the complement of the union of two real intervals, we represent the complex Green's function with the help of Jacobi's elliptic and theta functions. (iii) Using this representation, we explicitly obtain all parameters of the lemniscatic domain corresponding to the complement of the two intervals. In addition, using an equality between the corresponding complex Green's functions, we obtain a numerical method for computing the conformal map from the complement of two intervals onto a lemniscatic domain which yields more accurate results than a previous method from the literature.