c65da874-5ffc-4567-8041-cf98646eb06420241128091357172wseas:wseasmdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON MATHEMATICS2224-28801109-276910.37394/23206http://wseas.org/wseas/cms.action?id=40511220241220242310.37394/23206.2024.23https://wseas.com/journals/mathematics/2024.phpAndrei-FlorinAlbişoruFaculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, ROMANIADorinGhişaDepartment of Mathematics, Glendon College, York University, Toronto, CANADAThe 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.112820241128202480281483https://wseas.com/journals/mathematics/2024/b685106-042(2024).pdf10.37394/23206.2024.23.83https://wseas.com/journals/mathematics/2024/b685106-042(2024).pdf10.37394/23206.2023.22.72Albisoru, A.F. and Ghisa, D., Conformal Self-Mappings of the Fundamental Domains of Analytic Functions and Computer Experimentation, WSEAS Transactions on Mathematics, 22, 2023, 652-665. Granath, J., Finite Blaschke products and their properties, PhD Thesis, University of Stockholm, 2020. Ahlfors, L.V., Complex Analysis, International Series in Pure and Applied Mathematics, McGraw-Hill Book Company, New York, 1979. Cargo, G.T., The Boundary behavior of Blaschke Products, Journal of Mathematical Analysis and Applications, 5, 1-16, 1962. Garcia, S.R., Mashreghi, J. and Ross, W.T., Finite Blaschke products: a survey, Math and Computer Science Faculty Publications, 181, 1-30, 2018. 10.1080/17476930008815283Cassier, G. and Chalendar, I., The Group of the Invariants of a Finite Blaschke Product, Complex Variables, 42, 193-206, 2000. 10.1090/s0002-9939-1966-0193243-6Colwell, P., On the boundary behavior of Blaschke products in the unit disk, Proc. Amer. Math. Soc., 17, 582-587, 1966. 10.1090/s0002-9904-1939-07012-2Walsh, J.L., Note of Location of Zeros of the Derivative of a Rational Function whose Zeros and Poles Are Symmetric in a Circle, Bull. Amer. Math. Soc., 45, 462-470, 1939. Ahlfors, L.V. and Sario, L., Riemann Surfaces, Princeton University Press, Princeton, New Jersey, 1974. 10.1142/9789814313179_0001Andreian-Cazacu, C. and Ghisa, D., Global Mapping Properties of Analytic Functions, Proceedings of 7-th ISAAC Congress, London(U.K.), 3-12, 2009. Constantinescu, C. and Weber, K., in collaboration with Sontag, A., Integration Theory, Vol 1: Measure and Integral, Wiley-Interscience Publication, New York, 1985. Hardy, G.H. and Riesz, M., The General Theory of Dirichlet Series, Cambridge University Press, Cambridge, 1915.