METHOD OF SEARCHING COMPLEX SOLUTIONS OF INEQUALITIES BY THE DEFLECTION METHOD - Filier, Zalmen; Chuikov, Artem (Physical and Mathematical Education, 2021)
Formulation of the problem. Traditionally, the school textbooks deal with inequalities in the set of real numbers. Solving inequalities with the unknown number they are limited to finding the area where the requirement is greater or less is fulfilled. By the way, in a number of problems it is important how much the values are differs. At the same time, complex solutions are revealed in the case of a real deflection. Materials and methods. The methods of mathematical analysis and theory of functions of a complex variable are used. Analysis and modeling techniques are also used to develop algorithms for graphically representing our results in the computer mathematics system Maple 17. Results. It is proposed to use a complex deflection r = s + it, where s > 0 or s = 0 and t > 0, which gives complex solutions of inequalities. The set of all inequality solutions obtained by the complex residual method is a two-dimensional domain. Moreover, inequalities with opposite signs have solutions that complement each other to a complex plane. Examples of the application of the complex deflection method for solving quadratic, rational, and other inequalities are presented. The application of the computer mathematics system Maple 17 for graphical construction of the area-solutions of inequalities is demonstrated. Conclusions. The submitted material can be useful for school teachers and teachers of professional higher and higher education in studying the topic «Complex numbers». Inequalities in a complex set have been considered sporadically, for example, in proving D’Alembert’s lemma about the value of the modulus of a complex argument at adjacent points around a point where it is nonzero. These inequalities can be used to find the roots of complex functions. Further research in this area is to systematize and classify inequalities and methods of their solving in a complex plane.