Calculation of the value of the functions of the complex variable with by an interval argument, we will design in the hyperbolic form

Abstract

Information about the interval numbers presented in the classical form, the CENTER-RADIUS system and in the hyperbolic form is given. Rules for the transition from one of the forms of representation of interval numbers to others are proposed. Information is given on complex interval numbers, the real and imaginary parts of which are presented in hyperbolic form. The rules for performing basic arithmetic operations with these numbers and the calculation of interval values of power, exponential, logarithmic functions, direct and inverse trigonometric functions, direct and inverse hyperbolic functions are described. For functions of a complex variable, information about their real and imaginary parts is given. The list of functions corresponds to the functions of a complex variable included in the EXCEL system. Relationships are obtained for determining the real and imaginary parts of the secant, cosecant, tangent and cotangent functions for circular trigonometric and hyperbolic functions, which were absent in the most common reference literature. It is shown that the operations of multiplication, division and raising to an integer power are most appropriate to perform with complex interval numbers, which are defined in hyperbolic form. The operation of calculating the root of degree n from an interval complex number presented in hyperbolic form is most expediently performed using the CENTER-RADIUS system in combination with the hyperbolic form of representing the interval number. Relationships are obtained that make it possible to obtain a function of an interval complex variable equivalent to the original one and suitable for further work with complex functions and numbers presented in hyperbolic form and in the CENTER-RADIUS system. Examples illustrating the application of the proposed technique are given.