Complex Analytic Functions with Natural Boundary

The analytic functions with natural boundaries have been only occasionally mentioned in literature. They were defined mainly by lacunary power series of Hadamard type, except for the modular function which is the result of a laborious construction. The case of infinite Blaschke products which cannot be analytically continued over the unit circle is also known, yet the authors have no knowledge about any study devoted to these functions. The purpose of this article is to take a closer look upon these functions, to find new techniques of generating them and to bring this topic into the mainstream study of analytic functions. A special attention is devoted to the theory of Blaschke products, which is completed with new results related to their boundary behavior, making possible the study of the Blaschke products with natural boundary. We apply to them the same method of study as for ordinary infinite Blaschke products obtaining mirror functions with respect to the unit circle. The working tool is that of the fundamental domains, which are easily revealed by the technique of continuation over a curve, or lifting of a curve, having its origins in the differential geometry. Graphic illustrations contribute to a better understanding of the theoretical endeavors.