Objective: To develop an algorithm for the analysis of functions with asymptotes in the polar coordinate system and  apply it to the study of curves.

Theoretical Background: The study of curves has broad applications: in physics — for analyzing the trajectories of moving bodies; in astronomy — for examining the orbits of planets and satellites; in engineering — for designing aerodynamic shapes; and  in computer graphics — for modeling geometric objects.

Analyzing curves is an essential tool for understanding complex relationships. Methods of differential and integral calculus, as well as representing curves in various coordinate systems, allow researchers to identify key properties such as extrema, asymptotes, discontinuities, and other features. These approaches are widely applied across numerous fields of science and technology.

Method: The study employs methods of analysis and synthesis, as well as the mathematical apparatus of differential calculus. Particular attention is given to the investigation of a function in the polar coordinate system using derivatives to identify the key features of its behavior.

Results and Discussion: As a result of the study, an algorithm was developed for analyzing functions with asymptotes in the polar coordinate system, including cases involving discontinuities. The proposed method makes it possible to identify the key points of the curve without the use of computational tools, which makes it convenient for manual plotting in both polar and Cartesian coordinate systems. This approach provides a deeper understanding of the behavior of unbounded functions and expands the range of methods available for graphical analysis.

Research implications: A systematic approach to the study of curves in the polar coordinate system is presented, including the identification of oblique asymptotes and characteristic points without the use of computational software. This broadens the existing methodologies for function analysis and visualization.

Originality/Value: The work contributes to the development of analytical methods for studying curves defined in polar coordinates, offering both methodological and practical significance in engineering, physics, and mathematical disciplines.

Keywords: function, derivative, polar coordinates, curves, algorithm, asymptotes.