The First Mean Value Theorem of integral in Perplex Plane
DOI:
https://doi.org/10.54097/hak78696Keywords:
Perplex Number, The First Mean Value Theorem of Integral, Perplex Interval.Abstract
Perplex numbers are an extension of the real numbers, consisting of pairs of real numbers that form a commutative ring. This article considers the partial order structure of perplex numbers and discusses the concept of perplex interval. Based on the definition of integration of perplex functions of perplex variable, this article we investigate the first mean value theorem of integral in perplex plane which can significantly enrich the mathematical toolbox by providing essential methods for estimating and approximating perplex number functions. It also simplifies the solution of complex integral problems involving perplex numbers and fosters theoretical extensions by exploring the behavior of perplex number functions under various conditions. This, in turn, promotes the development of perplex calculus, enhancing the overall theoretical and practical framework of perplex number analysis. In physics, particularly in relativity, studying the first mean value theorem for perplex numbers can provide more precise methods for describing and estimating physical quantities, enhancing the application of hyperbolic numbers in these fields.
Downloads
References
[1] Fjelstad P. Extending special relativity via the perplex numbers [J]. American Journal of Physics, 1986, 54: 416-422.
[2] Sobczyk G. The Hyperbolic Number Plane [J]. The College Mathematics Journal, 1995, 26: 10.2307/2687027.
[3] Saini H., Sharma A., Kumar R. Some Fundamental Theorems of Functional Analysis with Bicomplex and Hyperbolic Scalars [J]. Advances in Applied Clifford Algebras, 2020, 30: 10.1007/s00006-020-01092-6.
[4] Kumar R., Saini H. Topological Bicomplex Modules [J]. Advances in Applied Clifford Algebras, 2016, 26: 10.1007/s00006-016-0646-1.
[5] Luna-Elizarrarás, M. & Shapiro, Michael & Struppa, Daniele & Vajiac, Adrian. (2015). Bicomplex Holomorphic Functions. 10.1007/978-3-319-24868-4.
[6] Balankin A.S., Bory-Reyes J., Luna-Elizarrarás M.E., Shapiro M. Cantor-type sets in hyperbolic numbers [J]. Fractals, 2016, 24(04).
[7] Luna-Elizarrarás M., Fangfang. Integration of Functions of a Hyperbolic Variable; Research on power load forecasting based on Improved BP neural network [J] [D]. Complex Analysis and Operator Theory; Harbin Institute of Technology, 2022; 16: 10.1007/s11785-022-01197-9, 2011.
[8] Tellez G., Bory-Reyes J. Extensions of the Shannon Entropy and the Chaos Game Algorithm to Hyperbolic Numbers Plane [J]. Fractals, 2021, 29: 1-15. DOI: 10.1142/S0218348X21500134.
[9] Soykan Y., Taşdemir E. A study on hyperbolic numbers with generalized Jacobsthal numbers components [J]. The International Journal of Nonlinear Analysis and Applications (IJNAA), 2022: 1-17. DOI: 10.22075/ijnaa.2021.22113.2328.
[10] Narita K., Endou N., Shidama Y. The First Mean Value Theorem for Integrals [J]. Formalized Mathematics, 2008, 16: 51-55. DOI: 10.2478/v10037-008-0008-0.
[11] Zhou Y. Mean Value Theorem and Its Uses [J]. Highlights in Science, Engineering and Technology, 2023, 72: 926-930. DOI: 10.54097/48pkqt70.
[12] Wu S. Proof and Application of the Mean Value Theorem [J]. Highlights in Science, Engineering and Technology, 2023, 72: 565-571. DOI: 10.54097/nw2nd028.
[13] Cakir H. Algebraic and geometric applications of hyperbolic numbers and hyperbolic number matrices [D]. Akdeniz University, 2017: 6.
[14] Nešović E., Petrović-Torgašev M. Some trigonometric relations in the Lorentzian plane [J]. Kragujevac J. Math, 2003, 33(25): 219-225.
[15] Öztürk İ., Özdemir M. Affine transformations of hyperbolic number plane [J]. Boletín de la Sociedad Matemática Mexicana, 2022, 28: 10.1007/s40590-022-00455-2.
[16] Emanuello J., Nolder C. Projective Compactification of R-1,R-1 and Its Mobius Geometry [J]. Complex Analysis and Operator Theory, 2015, 9: 329-354. DOI: 10.1007/s11785-014-0363-5.
[17] Agarwal R. Bicomplex Polygamma Function [J]. Tokyo Journal of Mathematics, 2007, 30: 10.3836/tjm/1202136693.
Downloads
Published
Issue
Section
License
Copyright (c) 2024 Highlights in Science, Engineering and Technology
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
