An introduction to the theory of cluster sets, a branch of topological analysis.

Most topics dealt with here deal with complex analysis of both one and several complex variables. Several contributions come from elasticity theory. Areas covered include the theory of p-adic analysis, mappings of bounded mean oscillations, quasiconformal mappings of Klein surfaces, complex dynamics of inverse functions of rational or transcendental entire functions, the nonlinear Riemann-Hilbert problem for analytic functions with nonsmooth target manifolds, the Carleman-Bers-Vekua system, the logarithmic derivative of meromorphic functions, G-lines, computing the number of points in an arbitrary finite semi-algebraic subset, linear differential operators, explicit solution of first and second order systems in bounded domains degenerating at the boundary, the Cauchy-Pompeiu representation in L2 space, strongly singular operators of Calderon-Zygmund type, quadrature solutions to initial and boundary-value problems, the Dirichlet problem, operator theory, tomography, elastic displacements and stresses, quantum chaos, and periodic wavelets.

For the first systematic investigations of the theory of cluster sets of analytic functions, we are indebted to IVERSEN [1-3J and GROSS [1-3J about forty years ago. Subsequent important contributions before 1940 were made by SEIDEL [1-2J, DOOE [1-4J, CARTWRIGHT [1-3J and BEURLING [1]. The investigations of SEIDEL and BEURLING gave great impetus and interest to Japanese mathematicians; beginning about 1940 some contributions were made to the theory by KUNUGUI [1-3J, IRIE [IJ, TOKI [IJ, TUMURA [1-2J, KAMETANI [1-4J, TsuJI [4J and NOSHIRO [1-4J. Recently, many noteworthy advances have been made by BAGEMIHL, SEIDEL, COLLINGWOOD, CARTWRIGHT, HERVE, LEHTO, LOHWATER, MEIER, OHTSUKA and many other mathematicians. The main purpose of this small book is to give a systematic account on the theory of cluster sets. Chapter I is devoted to some definitions and preliminary discussions. In Chapter II, we treat extensions of classical results on cluster sets to the case of single-valued analytic functions in a general plane domain whose boundary contains a compact set of essential singularities of capacity zero; it is well-known that HALLSTROM [2J and TsuJI [7J extended independently Nevanlinna's theory of meromorphic functions to the case of a compact set of essential singUlarities of logarithmic capacity zero. Here, Ahlfors' theory of covering surfaces plays a funda mental role. Chapter III "is concerned with functions meromorphic in the unit circle.