Calculation of the value of the functions of the complex variable with by an interval argument, we will design in the hyperbolic form
Article Sidebar
Main Article Content
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.
Article Details
References
Podzharenko, V.O, Vasilevskyi, O.M. and Kucheruk, V.Yu. (2008), Opratsiuvannia rezultativ vymiriuvan na osnovi kontseptsii nevyznachenosti: navchalnyi posibnyk [Developing measurement results based on the concept of uncertainty: a textbook], VNTU, Vynnytsa, 158 p.
Alefeld G,. and Herzberger, Yu. (1987), Vvedenie v interval'nye vychislenija.”[Introduction to interval calculations], Mir, Moscow, 360 p.
Zhukovska, O. A. (2009), Osnovy intervalnoho analizu: navchalnyi, [Fundamentals of interval analysis], Osvita Ukrainy, Kyiv, 136 p.
Dubnickij, V.Ju., Kobylin, A.M.,and Kobylin O.A. (2017), ”Vychislenie znachenij jelementarnyh i special'nyh funkcij s interval'no zadannym argumentom, opredeljonnym v sisteme Centr–Radius” [Calculation of the values of elementary and special functions with an interval -set argument defined in the Center–Radius system], Applied radio electronics, No. 3, pp. 147 -155.
Dubnickij, V.Ju., Kobylin, A.M. and Kobylin, O.A. (2019), “Calculation of similarity indicators and numerical integration of the criteria of similarity intervally defined in the Center-Radius system”, Advanced Information Systems, Vol. 3, No. 3, pp. 55-62, doi: https://doi.org/10.20998/2522-9052.2019.3.08.
Dubnitsky, V.Yu., Kobylin, A.M. and Kobylin, O.A (2010), “Intervalnoe ocenivanie parametrov dlja opredelenija nadjozhnosti programmnogo obespechenija” [Interval estimation of parameters to determine software reliability], Bionics of intelligence, No, 1 (72), pp. 43 -49.
Dubnytskyi, V.Yu., Kobylin, A.M. and Kobylin, O.A. (2020), Software system for operational evaluation of the interval efficiency of bank conversion operations intended for mobile devices on the Android operating system, computer program, Certificate of copyright registration for the work, No. 101306, December 17, 2020.
Kobylin, A.M., Dubnytskyi, V.Yu., Kobylin, O.A., Stiahlyk, N.I. and Vakulenko, N.O. (2021), A software system for operational assessment of the interval efficiency of currency operations, designed for mobile devices, Certificate of copyright registration for the work, No. 108729, October 20, 2021.
Svezhenceva, O. V. and Umnova, M. O. (2015), “Raschet ustanovivshihsja rezhimov radial'noj jelektricheskoj seti na naprjazhenii 0,4 kv interval'nym metodom” [Calculation of steady-state regimes of a radial electrical network at a voltage of 0.4 kV by the interval method], Bulletin of the Irkutsk State Technical Universit, No. 3 (98). pp. 215-222.
Kryukov, A V. and Litvintsev, A,I, (2013), “Interval'noe modelirovanie avarijnyh rezhimov jelektrojenergeticheskih system” [Interval simulation of emergency modes of electric power systems], Systems Methods Technologies, No. 4 (20), pp, 73-79.
Hadetska, S.V., Dubnytskyi, V.Yu., Kushneruk, Yu.I., and Khodyriev, O.I. (2022), Performing basic arithmetic operations with complex numbers, which are presented in interval hyperbolic form, Advanced Information Systems, Vol. 6, No. 1, pp. 104-113, doi: https://doi.org/10.20998/2522-9052.2022.1.17.
Kalmykov, S.A., Shokin, Ju.I. and Juldashev, 3.X. (1986), Metody interval'nogo analiza [Methods of interval analysis], Nauka, Novosibirsk, 1986. 223 p.
Sharyj, S.P. (2012), Konechnomernyj interval'nyj analiz [Finite-Dimensional Interval Analysis], XYZ, Novosibirsk, 606 p.
Dubnitsky, V.Yu., Kobilin, A.M., Kobilin, O.A. and Kushneruk, Yu.I.n(2021), “EXCEL-oriented procedure for calculating the values of special functions with an interval argument, given in hyperbolic form.”, Advanced Information Systems, Vol. 5, No. 4, pp. 116-123, doi: https://doi.org/10.20998/2522-9052.2021.4.
Molodtsov, D. A. and Kovkov, D. V. (2011), “Vvedenie v teoriju priblizhennyh chisel” [Introduction to the theory of approximate numbers], Bulletin of the Tver State University. Series: Applied Mathematics, No. 23, pp. 111-128.
Spivak, I. Ya. And Krepych, I. Ya. A, (2019), “Prykladni aspekty intervalnykh obchyslen: navchalnyi posibnyk” [Applied aspects of interval calculations: a study guide], Palanytsia FOP, Ternopil, 153 p.
Kononyuk, A. E. (2011), “Dyskretno-nepreryvnaia matematyka. Alhebry. K.4.Ch.2.“[Discrete-continuous mathematics. Algebras. K.4.Part 2], Education of Ukraine, Kyiv, 668 p.
Smirnova, E. N. and Maksimenko N. V. (2015), Jelementy interval'nogo analiza: metodicheskie ukazanija [Elements of interval analysis: guidelines], Orenburg State University, Orenburg, 62 p,
Verbitsky, I. L., and Fainberg, E. D. (1989), Jelementy teorii funkcij kompleksnogo peremennogo [Elements of the theory of functions of a complex variable], UZPI, Kharkiv,135 p.
Serogodsky, V. V., Finkov, M. V., Kozlov, D. A., Prokdi, R. G. et al. (2017), Excel 2016. Polnoe rukovodstvo [Excel 2016, Complete guide], Nauka i tehnika, 412 p.
Korn, G., and Korn, T. (1968), “Spravochnik po matematike dlya nauchnykh rabotnikov i inzhenerov” [Handbook of mathematics for scientists and engineers], Science, Moscow, 720p,
Bronstein, I.N. and Semendyaev, K.A. (1981), “Spravochnik po matematike dlya inzhenerov i uchashchikhsya vtuzov” [Handbook of mathematics for engineers and students of higher educational institutions], Nauka, TPH, Moscow, 719 p.