Boundary lines of analytic functions with a parameter

The study of various processes (electric and magnetic fields, the flow of air and liquids, quantum physics, etc.) comes down to the study of analytical functions of a complex variable. The subject of research is the analytic functions of a complex variable with a parameter. The research introduces the concepts of boundary, regular, and singular points for such classes of functions and defines the following concepts: singular, regular, and boundary regions. Using the topological approach, the study broadens the concept of a boundary line. The study selects the main boundary line from the set of boundary lines. The examples illustrate the topology and various forms of the main boundary lines (closed, spider-like, with a punctured point, a countable amount) as well as the structure of the boundary area. As shown in the examples throughout, it is impossible to determine the shape of the main boundary lines for the general case, therefore each case must be considered separately.